we have $\Theta(\sigma(\phi(\xi)))\in
W^{uu}_{g_{2},R}(\Theta(\phi(\sigma(\xi))))$. Indeed, recall that we had fixed
$\mathcal{R}_{g}=\\{R_{g,1},\cdots,R_{g,m}\\}$ a small Markov partition for
$g_{2}$. Since $\phi(\xi)\in\Sigma^{u}_{loc}(\xi)$, we obtain that
$\Theta(\phi(\xi))$ and $\Theta(\xi)$ belongs to the same local unstable
manifold intersected with some rectangle $R_{g,i}$. Since the expansion rate
of unstable manifolds for $g_{2}$ is close to $\lambda^{-2N}$, which is a
constant, there exists $R_{1}>0$ that verifies $\Theta(\sigma(\phi(\xi)))\in
W^{uu}_{g_{2},R_{1}}(\Theta(\sigma(\xi)))$, for any $\xi\in\Sigma$. To
conclude, we observe that $\Theta(\phi(\sigma(\xi)))\in
W^{uu}_{g_{2},loc}(\Theta(\sigma(\xi)))$. Hence, by fixing $R$ sufficiently
large we conclude our claim.

Since $g$ is $C^{2+\alpha}$, Theorem 2.10 in the appendix implies that for
every $\xi\in\Sigma$, the holonomy
$H^{u}_{\sigma(\phi(\xi)),\phi(\sigma(\xi))}$ is a $C^{2}$-diffeomorphism of
$\mathbb{T}^{2}$ with uniformly bounded $C^{2}$-norm. Since
$\hat{g}_{\xi}=g_{\Theta(\xi)}$, we also have that all the
$C^{2}$-diffeomophisms $\hat{g}_{\xi}$ belong to a compact subset of
$\mathrm{Diff}^{2}(\mathbb{T}^{2})$. We conclude that for every $\xi$, the
$C^{2}$-norm of $\tilde{g}_{\xi}$ is uniformly bounded. Similar conclusion
holds for $\tilde{g}^{-1}_{\xi}$. In particular, the skew product $\tilde{g}$
verifies the integrability condition (22).

It is easy to see that the fiber-wise Lyapunov exponents of
$(\tilde{g},\tilde{\mu})$ are the same as the center Lyapunov exponents of
$(g,\mu)$. In particular, $\tilde{\mu}$ is a hyperbolic measure with a
positive and a negative fiber-wise Lyapunov exponent.

Lemma 6.4 states that $(\tilde{g},\tilde{\mu})$ verifies the conditions $(1)$
and $(2)$ in the hypothesis of Theorem 6.2. Since the skew products
$\tilde{g}$ fibers over the system $(\sigma,\nu_{\sigma})$, which is ergodic,
we conclude that either

  1. 1.

the measure $\tilde{\mu}_{\xi}$ is atomic for $\nu_{\sigma}$-almost every
$\xi$;

  2. 2.

$\tilde{\mu}$ is fiber-wise SRB;

  3. 3.

the stable distribution $(x,\xi)\mapsto E^{-}_{\tilde{g}}(x,\xi)$ is
$\mathcal{B}^{*}$-measurable.

Notice that the composition $(\Phi\circ\Psi)$ takes fibers of
$\mathbb{T}^{2}\times\mathbb{T}^{2}$ into fibers of
$\mathbb{T}^{2}\times\Sigma$. Furthermore, it acts as a $C^{2}$-diffeomorphism
on each fiber. Observe also that it measurably conjugates the dynamics of $g$
and $\tilde{g}$ on a set of full $\mu$-measure. In particular, for
$\nu$-almost every $p_{2}\in\mathbb{T}^{2}$ we have

$\mu^{c}_{p_{2}}=(\Phi\circ\Psi)^{-1}_{*}\tilde{\mu}_{\Theta^{-1}(p_{2})}.$
(26)

From (26) above, $\tilde{\mu}_{\xi}$ is atomic if and only if
$\mu_{\Theta(\xi)}$ is atomic, for $\nu_{\sigma}$-almost every $\xi$.

Since $\mu$ is an $u$-Gibbs measure, it will be an SRB measure if and only if
it is fiber-wise SRB in the sense of definition 6.1. From (26), we conclude
that $\tilde{\mu}$ is fiber-wise SRB for $\tilde{g}$ if and only if $\mu$ is
fiber-wise SRB for $g$.

For the map $(x,\xi)\mapsto E^{-}_{\tilde{g}}(x,\xi)$ to be
$\mathcal{B}^{*}$-measurable, it is equivalent to the following: for
$\tilde{\mu}$-almost every $(x,\xi)$ and for $\nu_{\sigma}^{\xi}$-almost every
$\eta\in\Sigma^{u}_{loc}(\xi)$, we have that
$E^{-}_{\tilde{g}}(x,\xi)=E^{-}_{\tilde{g}}(x,\eta)$. Observe that the points
$(x,\xi)$ and $(x,\eta)$ belong to the same local unstable set for
$\tilde{g}$. By the conjugacy $(\Psi\circ\Phi)$, we conclude that

$E^{-}_{\tilde{g}}(x,\xi)=DH^{u}_{\Theta(\xi),\Theta(\phi(\xi))}(x)E^{-}_{g,(x,\Theta(\xi))}.$

Since the measure is $u$-Gibbs, the third condition above is equivalent to for
$\mu$-almost every $p\in\mathbb{T}^{4}$, for Lebesgue almost every point $q\in
W^{uu}_{loc}(p)$, we have $E^{-}_{g,q}=DH^{u}_{p,q}(p)E^{-}_{g,p}$.

All these conclusions hold for any
$g\in\mathrm{Diff}^{2+\alpha}(\mathbb{T}^{2})$ sufficiently $C^{2}$-close to
$f_{N}$. This concludes the proof. ∎

We remark that the same proof of Theorem 6.3 also gives the following theorem.

######  Theorem 6.5.

Let $S$ be a compact surface and let $\alpha\in(0,1)$ be a constant. Suppose
that $g\in\mathrm{Sk}^{2+\alpha}(S\times\mathbb{T}^{2})$ be a partially
hyperbolic skew product which is $(2,\alpha)$-unstable center bunched. If
$\mu\in\mathrm{Gibbs}^{u}(g)$ is an ergodic measure with one positive and one
negative exponent along the center, then either

  1. 1.

$\mu$ is an SRB measure;

  2. 2.

for $\mu$-almost every $p$ and for Lebesgue almost every point $q\in
W^{uu}_{loc}(p)$,

$E^{-}_{g,q}=DH^{u}_{p,q}(p)E^{-}_{g,p};$

  3. 3.

for $\mu$-almost every $p$, the measure $\mu^{c}_{p}$ is atomic.

##  7 The non invariance of stable directions by $u$-holonomies

In this section we fix $N$ large and $\mathcal{U}_{N}$ small enough such that
Theorem 4.1 holds for some small fixed $\delta>0$. In particular, if
$g\in\mathcal{U}_{N}$ then any $u$-Gibbs measure for $g$ has both a positive
and a negative center Lyapunov exponent for $\mu$ almost every point. Since
$\mu$ has absolutely continuous disintegration with respect to strong unstable
manifolds, for $\mu$-almost every point $p$, Lesbesgue almost every point
$q\in W^{uu}_{g}(p)$ has a well defined Oseledec’s stable and unstable
directions in the center, where the Lebesgue measure we are considering is the
measure restricted to the strong unstable manifold $W^{uu}_{g}(p)$.

Recall that for any $p\in\mathbb{T}^{4}$ and any $q\in W^{uu}_{g}(p)$, there
is a well defined unstable holonomy map $H^{u}_{p,q}:W^{c}_{g}(p)\to
W^{c}_{g}(q)$. Furthermore, this map is a $C^{1}$-diffeomorphism. The main
result in this section is the following:

######  Proposition 7.1.

Let $g\in\mathcal{U}_{N}$ and let $\mu$ be an $u$-Gibbs measure for $g$. For
any $\varepsilon>0$, the following property holds: for $\mu$-almost every $p$,
there exists a set $D^{u}$ contained in $W^{uu}_{g,\varepsilon}(p)$ with
positive Lebesgue measure (for the riemannian volume of
$W^{uu}_{g,\varepsilon}(p)$) such that for any $q\in D^{u}$ it is verified
that

$DH^{u}_{p,q}(p)E^{-}_{g,p}\neq E^{-}_{g,q}.$

The rest of this section is dedicated to prove proposition 7.1.

Let $g\in\mathcal{U}_{N}$, for any $p\in\mathbb{T}^{4}$, for any piece of
strong unstable manifold $\gamma^{u}_{p}$ containing $p$ and any unit vector
$v\in E^{c}_{g,p}$, we define a unitary vector field over $\gamma^{u}_{p}$
defined as follows: for any $q\in\gamma^{u}_{p}$ we write

$PH^{u}_{p,q}(p)v=\displaystyle\frac{DH^{u}_{p,q}(p)v}{\|DH^{u}_{p,q}(p)v\|},$
(27)

and define $v^{\prime}_{q}:=PH^{u}_{p,q}(p)v$. First we study the regularity
of the vector field $v^{\prime}$.

######  Lemma 7.2.

Let $g\in\mathcal{U}_{N}$. There exists a constant $C>0$ that verifies the
following: for any $p\in\mathbb{T}^{4}$, let $\gamma^{u}_{p}:=W^{uu}_{g,1}(p)$
be the strong unstable manifold of size $1$, for any unit vector $v\in
E^{c}_{g,p}$, the vector field $v^{u}$ defined above is
$(C,\frac{1}{2})$-Hölder.

###### Proof.

Observe that, for $N$ large enough, we have

$\left(\lambda^{2N}\right)^{\frac{1}{2}}<(4N^{2})^{-1}\textrm{ and
}\left(\lambda^{2N}\right)^{\frac{1}{2}}<(2N)^{-1}.$

This means that $f_{N}$ verifies the conditions (1) and (2) from Theorem 2.8,
for $\theta=\frac{1}{2}$. In particular, any $g$ sufficiently $C^{1}$-close to
$f_{N}$ also verifies (1) and (2). Lemma 7.2 then follows from the conclusion
(3), for unstable holonomies, of Theorem 2.8. ∎

Next, we will see how the center bunching condition “smoothes” a center vector
field over a piece of strong unstable manifold. This is a crucial point for
us, so that it will allow us to apply some of the techniques and estimates
from section 4 to prove proposition 7.1.

######  Lemma 7.3.

Let $g\in\mathcal{U}_{N}$. For any piece of strong unstable curve $\gamma^{u}$
and any $X$ unitary vector field over $\gamma^{u}$ tangent to $E^{c}_{g}$
which is $(C_{0},\frac{1}{2})$-Hölder, for some $C_{0}:=C_{0}(X)>0$, the
following holds: there exists $n_{0}\in\mathbb{N}$, which depends only on
$C_{0}$, such that for every $n\geq n_{0}$, the vector field
$X_{n}:=\frac{g^{n}_{*}(X)}{\|g^{n}_{*}(X)\|}$ over $g^{n}(\gamma^{u})$ is
$(C_{n},\frac{1}{2})$-Hölder with $C_{n}<30N^{2}\lambda^{N}$.

###### Proof.

The proof of this lemma is essentially contained in the proof of Lemma $1$
from [10]. However, we will repeat the main steps of the argument here. For
simplicity we will prove the lemma for $f_{N}$, which we will denote by $f$.
Using the estimates from Lemma 4.4, one can adapt the calculations for any
$g\in\mathcal{U}_{N}$.

Let us just review some estimates for $f$. Recall that

$Df(x,y,z,w)=\begin{pmatrix}Ds_{N}(x,y)&P_{x}\circ A^{N}(z,w)\\\
0&A^{2N}(z,w)\end{pmatrix}.$

Hence, $\|Df(x,y,z,w)|_{E^{c}}\|=\|Ds_{n}(x,y)\|\leq 2N$. Since $Ds_{N}$ is
the only non linear term, $\|D^{2}f\|=\|D^{2}s_{N}\|\leq N$.

Let $\gamma^{u}$ be a piece of a strong unstable manifold and $X$ a
$(C_{0},\frac{1}{2})$-Hölder unitary vector field over $\gamma^{u}$. Let us
estimate $C_{1}$, the Hölder constant of $X_{1}$ over $f(\gamma^{u})$. First,
for any $m,m^{\prime}\in\gamma^{u}$, we have

$\begin{array}[]{l}\|Df(m)X_{m}-Df(m^{\prime})X_{m^{\prime}}\|\leq\\\
\|Df(m)X_{m}-Df(m)X_{m^{\prime}}\|+\|Df(m)X_{m^{\prime}}-Df(m^{\prime})X_{m^{\prime}}\|=I+II.\end{array}$

Since $X$ is $(C_{0},\frac{1}{2})$-Hölder, we obtain

$I\leq 2N\|X_{m}-X_{m^{\prime}}\|\leq 2NC_{0}d(m,m^{\prime})^{\frac{1}{2}}.$

If $d(m,m^{\prime})\leq 1$, then

$II\leq
Nd(m,m^{\prime})<Nd(m,m^{\prime})^{\frac{1}{2}}<7Nd(m,m^{\prime})^{\frac{1}{2}}.$

Observe that $d(m,m^{\prime})\leq 2\pi<7$, for any two points
$m,m^{\prime}\in\mathbb{T}^{4}$. If $d(m,m^{\prime})>1$, then

$II\leq Nd(m,m^{\prime})\leq 7N<7Nd(m,m^{\prime})^{\frac{1}{2}}.$

We conclude that

$\|Df(m)X_{m}-Df(m^{\prime})X_{m^{\prime}}\|<(7N+2NC_{0})d(m,m^{\prime})^{\frac{1}{2}}.$
(28)

Also,

$\begin{array}[]{rcl}\|(X_{1})_{m}-(X_{1})_{m^{\prime}}\|&=&\displaystyle\frac{1}{\|f_{*}X_{m}\|\|f_{*}X_{m^{\prime}}\|}\left\|\|f_{*}X_{m^{\prime}}\|f_{*}X_{m}-\|f_{*}X_{m}\|f_{*}X_{m^{\prime}}\right\|\\\
&\leq&\displaystyle\frac{1}{\|f_{*}X_{m}\|\|f_{*}X_{m^{\prime}}\|}\left(\left\|\|f_{*}X_{m^{\prime}}\|f_{*}X_{m}-\|f_{*}X_{m^{\prime}}\|f_{*}X_{m^{\prime}}\right\|\right.\\\
&&\displaystyle+\left.\left\|\|f_{*}X_{m^{\prime}}\|f_{*}X_{m^{\prime}}-\|f_{*}X_{m}\|f_{*}X_{m^{\prime}}\right\|\right)\\\
&\leq&\displaystyle\frac{2}{\|f_{*}X_{m}\|}\|f_{*}X_{m}-f_{*}X_{m^{\prime}}\|\\\
&=&\displaystyle\frac{2}{\|f_{*}X_{m}\|}\|Df(f^{-1}(m))X_{f^{-1}(m)}-Df(f^{-1}(m^{\prime}))X_{f^{-1}(m^{\prime})}\|.\end{array}$

Using (28) for the points $f^{-1}(m)$ and $f^{-1}(m^{\prime})$, we have

$\|Df(f^{-1}(m))X_{f^{-1}(m)}-Df(f^{-1}(m^{\prime}))X_{f^{-1}(m^{\prime})}\|\leq
N\lambda^{N}(1+\lambda^{N})(7+2C_{0})d(m,m^{\prime})^{\frac{1}{2}}.$

Recall that $\|Df|_{E^{c}_{f}}\|\geq(2N)^{-1}$, hence

$\|X_{1}(m)-X_{1}(m^{\prime})\|\leq\displaystyle\frac{2N\lambda^{N}(1+\lambda^{N})(7+2C_{0})d(m,m^{\prime})^{\frac{1}{2}}}{\|f_{*}X(m)\|}\leq
4(1+\lambda^{N})N^{2}\lambda^{N}(7+2C_{0})d(m,m^{\prime})^{\frac{1}{2}}.$

Observe that $4(1+\lambda^{N})N^{2}\lambda^{N}(7+2C_{0})$ estimates the Hölder
constant of $X_{1}$. If $C_{0}\leq\frac{1}{10}$, then for $N$ large enough

$\displaystyle 4(1+\lambda^{N})N^{2}\lambda^{N}(7+2C_{0})\leq
4(1+\lambda^{N})N^{2}\lambda^{N}(7.2)<30N^{2}\lambda^{N}.$

Hence, $C_{1}<30N^{2}\lambda^{N}$ and the same calculations imply that
$C_{n}<30N^{2}\lambda^{N}$, for every $n\geq 1$. Now suppose that
$C_{0}>\frac{1}{10}$. Then, for $N$ large enough

$\displaystyle\frac{4(1+\lambda^{N})N^{2}\lambda^{N}(7+2C_{0})}{C_{0}}=4(1+\lambda^{N})N^{2}\lambda^{N}\left(\frac{7}{C_{0}}+2\right)<4(1+\lambda^{N})N^{2}\lambda^{N}72<\frac{1}{2}.$

This implies that $C_{1}<\frac{C_{0}}{2}$. Therefore, there exists
$\tilde{n}\in\mathbb{N}$ such that
$C_{\tilde{n}}<\left(\frac{1}{2}\right)^{\tilde{n}}C_{0}\leq\frac{1}{10}$.
Take $n_{0}=\tilde{n}+1$. We conclude that for every $n\geq n_{0}$,
$C_{n}<30N^{2}\lambda^{N}$. ∎

###### Proof of proposition 7.1.

If the conclusion of proposition 7.1 did not hold, there would exist a
diffeomorphism $g\in\mathcal{U}_{N}$, an $u$-Gibbs measure $\mu$ and a
measurable set $D$ of positive $\mu$-measure such that for any $p\in D$ and
for Lebesgue almost every point $q\in W^{uu}_{g}(p)$ we would have
$DH^{u}_{p,q}(E^{-}_{g,p})=E^{-}_{g,q}$. Fix $p\in D$ and let
$\gamma^{u}:=W^{uu}_{g,1}(p)$. Consider $v$ an unit vector on $E^{-}_{g,p}$
and let $v^{\prime}$ be the unit vector field over $\gamma^{u}$ defined as in
(27).

Let $C$ be the constant given by lemma 7.2. Therefore, $v^{\prime}$ is a
$(C,\frac{1}{2})$-Hölder vector field over $\gamma^{u}$. Let
$n_{0}\in\mathbb{N}$ be given by lemma 7.3. Hence, for $n\geq n_{0}$, the
vector field
$v_{n}^{\prime}:=\frac{g^{n}_{*}(v^{\prime})}{\|g^{n}_{*}(v^{\prime})\|}$ is
$(C_{n},\frac{1}{2})$-Hölder over $\gamma^{u}_{n}:=g^{n}(\gamma^{u})$, with
$C_{n}<30N^{2}\lambda^{N}$.

Suppose that $n_{0}$ is large enough such that $l(\gamma^{u}_{n_{0}})>2\pi$.
Hence, we may consider a $C^{1}$-curve
$\tilde{\gamma}:[0,2\pi]\to\mathbb{T}^{4}$ such that
$\tilde{\gamma}=(\tilde{\gamma}_{x},\tilde{\gamma}_{y},\tilde{\gamma}_{z},\tilde{\gamma}_{w})$
with $\left|\frac{d\tilde{\gamma}_{x}}{dt}\right|=1$,
$\tilde{\gamma}([0,2\pi])\subset\gamma^{u}_{n_{0}}$, and define
$\tilde{v}=v^{u}_{n_{0}}$. Following definition 4.9, the pair
$(\tilde{\gamma},\tilde{v})$ is an adapted field.

Recall that $\delta>0$ is fixed and in section 4, on the proof of Theorem 4.1,
we fixed $\tilde{\delta}=\frac{2\delta}{15}$. For each $k\geq 0$, we write
$\tilde{v}_{k}=v^{\prime}_{n_{0}+k}$ and recall that there exists
$N_{k}\in\mathbb{N}$ such that

$g^{k}\circ\tilde{\gamma}=\tilde{\gamma}^{k}_{1}\ast\cdots\ast\tilde{\gamma}^{k}_{N_{k}}\ast\tilde{\gamma}^{k}_{N_{k}+1},$

where $\tilde{\gamma}^{k}_{j}$ is an $u$-curve for $j=1,\cdots,N_{k}$ and
$\tilde{\gamma}^{k}_{N_{k}+1}$ is a segment of a $u$-curve. By lemma 4.13,
every pair $(\tilde{\gamma}^{k}_{j},\tilde{v}_{k}|_{\tilde{\gamma}^{k}_{j}})$
is an adapted field for $j=1,\cdots,N_{k}$.

Recall that in section 4, we had defined the notion of $\tilde{\delta}$-good
adapted field (see definition 4.12). We will need the following lemma.

######  Lemma 7.4 ([45], Lemma $7.27$).

Let $g\in\mathcal{U}_{N}$, and let $(\gamma,X)$ be a $\tilde{\delta}$-bad
adapted field. Then there exists a strip $S$ of length $\pi$ such that for
every $j$ satisfying $g^{-1}\gamma_{j}^{1}\subset S$, the field
$(\gamma_{1}^{j},\frac{g_{*}X}{\|g_{*}X\|})$ is $\tilde{\delta}$-good.

Let $(\hat{\gamma},\hat{v})$ be a $\tilde{\delta}$-good adapted field defined
as follows: if $(\tilde{\gamma},\tilde{v})$ is a $\tilde{\delta}$-good adapted
field then $(\hat{\gamma},\hat{v})=(\tilde{\gamma},\tilde{v})$. Otherwise, by
the previous lemma, we may choose $j\in\\{1,\cdots N_{k}\\}$ such that
$(\tilde{\gamma}^{1}_{j},\tilde{v}_{1}|_{\tilde{\gamma}^{1}_{j}})$ is a
$\tilde{\delta}$-good adapted field. In this case, we define
$(\hat{\gamma},\hat{v})=(\tilde{\gamma}^{1}_{j},\tilde{v}_{1}|_{\tilde{\gamma}^{1}_{j}})$.

Let $K\in\\{n_{0},n_{0}+1\\}$ be such that
$g^{-K}(\hat{\gamma})\subset\gamma^{u}$ and write
$\hat{\gamma}_{-K}:=g^{-K}(\hat{\gamma})$. Recall that we had defined
$J^{uu}_{g^{k}}(.)=|\det Dg^{k}(.)|_{E^{uu}_{g}}|$. For any $n\in\mathbb{N}$,

$\begin{array}[]{rcl}\displaystyle\frac{1}{|\hat{\gamma}|}\int_{\hat{\gamma}_{-K}}\log\|Dg^{K+n}v^{\prime}\|d\hat{\gamma}_{-K}&=&\displaystyle\frac{1}{|\hat{\gamma}|}\left(\sum_{i=0}^{K-1}\int_{g^{i}\circ\hat{\gamma}_{-K}}\log\|Dgv^{\prime}_{i}\|J^{uu}_{g^{-i}}d(g\circ\hat{\gamma}_{-K})\right.\\\
&&\displaystyle\left.+\int_{\hat{\gamma}}\log\|Dg^{n}\hat{v}\|J^{uu}_{g^{-K}}d\hat{\gamma}\right)\\\
&=&\displaystyle
M_{K}+\frac{1}{|\hat{\gamma}|}\int_{\hat{\gamma}}\log\|Dg^{n}\hat{v}\|J^{uu}_{g^{-K}}d\hat{\gamma}=M_{k}+I^{\hat{\gamma},\hat{v}}_{n},\end{array}$

where $M_{K}$ does not depend on $n$. Since $(\hat{\gamma},\hat{v})$ is a
$\tilde{\delta}$-good curve, by (19) in section 4, for $n$ large enough we
have

$\frac{I^{\hat{\gamma},\hat{v}}_{n}}{n}\geq(1-14\tilde{\delta})\log N.$

Therefore,

$\displaystyle\limsup_{n\to+\infty}\frac{1}{|\hat{\gamma}|}\int_{\hat{\gamma}_{-K}}\frac{\log\|Dg^{K+n}v^{u}\|}{n}d\hat{\gamma}_{-K}=\limsup_{n\to+\infty}\frac{M_{K}}{n}+\frac{I^{\hat{\gamma},\hat{v}}_{n}}{n}\geq(1-14\tilde{\delta})\log
N>0.$ (29)

However, by assumption, for Lebesgue almost every $q\in W^{uu}_{g}(p)$ the
vector $v^{u}_{q}$ belongs to $E^{-}_{g,q}$. In particular, there exists a
number $\lambda^{-}<0$ such that for Lebesgue almost every $q\in
W^{uu}_{g}(p)$

$\displaystyle\lim_{n\to+\infty}\frac{\log\|Dg^{K+n}(q)v^{u}_{q}\|}{n}=\lambda^{-}<0.$
(30)

By (30) and applying the dominated convergence theorem, we obtain

$\begin{array}[]{rcl}\displaystyle\limsup_{n\to+\infty}\frac{1}{|\hat{\gamma}|}\int_{\hat{\gamma}_{-K}}\frac{\log\|Dg^{K+n}v^{u}\|}{n}d\hat{\gamma}_{-K}&=&\displaystyle\frac{1}{|\hat{\gamma}|}\int_{\hat{\gamma}_{-K}}\limsup_{n\to+\infty}\frac{\log\|Dg^{K+n}v^{u}\|}{n}d\hat{\gamma}_{-K},\\\
&=&\displaystyle\frac{|\hat{\gamma}_{-K}|}{|\hat{\gamma}|}\lambda^{-}<0.\end{array}$

which is a contradiction with (29). ∎

##  8 Measures with atomic center disintegration and the proof of Theorem B

In this section we conclude the proof of Theorem B. The main ingredient that
is missing to conclude this proof is the following theorem.

######  Theorem 8.1.

Let $g\in\mathrm{Sk}^{2}(\mathbb{T}^{2}\times\mathbb{T}^{2})$ be a partially
hyperbolic, center bunched skew product. Suppose that there exists a constant
$\theta\in(0,1)$ such that

$E^{uu}_{g}\textrm{ is $\theta$-H\"{o}lder and
}\|Dg|_{E^{ss}_{g}}\|^{\theta}<m(Dg|_{E_{g}^{c}}).$

Let $\mu$ be an ergodic $u$-Gibbs measure for $g$ that verifies:

  1. 1.

$\mu$ has atomic center disintegration;

  2. 2.

for $\mu$-almost every $p\in\mathbb{T}^{4}$, and for Lebesgue almost every
point $q$ in $W^{uu}_{loc}(p)$

$E^{-}_{g,q}\neq DH^{u}_{p,q}(p)E^{-}_{g,p}.$

Then there exists a finite number of $C^{1}$ two dimensional tori
$T^{1}_{\mu},\cdots,T^{l}_{\mu}$ such that each of them is tangent to
$E^{ss}_{g}\oplus E^{uu}_{g}$ and
$\mathrm{supp}(\mu)=\cup_{i=1}^{l}T^{i}_{\mu}$.

######  Remark 8.2.

Theorem 8.1 also holds for a partially hyperbolic skew product
$g\in\mathrm{Sk}^{2}(S\times\mathbb{T}^{2})$, where $S$ is a compact surface,
verifying the rest of the hypotheses of the theorem.

In section 8.2, we reduce the proof of Theorem 8.1 into proving the
$s$-invariance of the center conditional measures, given by Theorem 8.3. The
proof of Theorem 8.3 is then given in section 9.

###  8.1 Proof of Theorem B assuming Theorem 8.1

Let $\alpha\in(0,1)$ and take $N$ large enough such that Theorem 6.3 holds and
let $\mathcal{U}_{N}^{sk}$ be a small $C^{2}$-neighborhood of $f_{N}$ is
$\mathrm{Sk}^{2}(\mathbb{T}^{2}\times\mathbb{T}^{2})$. Take
$g\in\mathcal{U}_{N}^{sk}\cap\mathrm{Sk}^{2+\alpha}(\mathbb{T}^{4})$ and take
an ergodic measure $\mu\in\mathrm{Gibbs}^{u}(g)$. By Theorem 6.3, there are
three possibilities:

  1. 1.

for $\mu$-almost every $p\in\mathbb{T}^{4}$, the measure $\mu^{c}_{p}$ is
atomic;

  2. 2.

$\mu$ is SRB;

  3. 3.

for $\mu$-almost every $p\in\mathbb{T}^{4}$, for Lebesgue almost every point
$q$ in $W^{uu}_{loc}(p)$, we have

$E^{-}_{g,q}=DH^{u}_{p,q}(p)E^{-}_{g,p}.$

Let us verify that $g$ verifies the Hölder condition in the statement of
Theorem 8.1. If $\theta\in(0,1)$ is a number that verifies

$\displaystyle\frac{\|Dg(p)|_{E^{c}_{g}}\|}{m(Dg(p)|_{E^{uu}_{g}})}<m(Dg(p)|_{E^{ss}_{g}})^{\theta},\textrm{
for all $p\in\mathbb{T}^{4}$,}$

then $E^{uu}_{g}$ is $\theta$-Hölder (see Section $4$ from [50]). In
particular, the maximum $\theta$ we can take is arbitrarily close to

$\displaystyle\inf_{p\in\mathbb{T}^{4}}\left\\{\frac{\log
m(Dg(p)|_{E^{uu}_{g}})-\log\|Dg(p)|_{E^{c}_{g}}\|}{-\log
m(Dg(p)|_{E^{ss}_{g}})}\right\\}.$ (31)

From the estimates of Lemma 4.4, for some small $\varepsilon_{1}>0$, if $N$ is
sufficiently large, we have that (31) is greater than

$\displaystyle\frac{2N\log\tilde{\mu}\left(1-\frac{(\varepsilon_{1}+\log
2N)}{2N\log\tilde{\mu}}\right)}{2N\log\tilde{\mu}\left(1-\frac{\varepsilon_{1}}{2N\log\tilde{\mu}}\right)},$

and this can be made arbitrarily close to $1$ by taking $N$ sufficiently large
(we remark that an analogous estimate of item $1$ from Lemma 4.4 holds for the
strong stable direction).

On the other hand, we also want
$\|Dg|_{E^{ss}_{g}}\|^{\theta}<m(Dg|_{E^{c}_{g}})$. From the estimates of
Lemma 4.4 for this inequality to hold we need

$\theta>\displaystyle\frac{\log
2N}{2N\log\tilde{\mu}\left(1-\frac{\varepsilon_{1}}{2N\log\tilde{\mu}}\right)}.$

The right side of the inequality goes to $0$ as $N$ increases. So for $N$
sufficiently large, we may take $\theta\in(0,1)$ that verifies the hypothesis
of Theorem 8.1.

Let $\mu$ be an ergodic $u$-Gibbs measure for $g$. By Proposition 7.1, $\mu$
cannot verify item $3$ above. If $\mu$ has atomic center disintegration, then
$\mu$ verifies the hypothesis of Theorem 8.1, and therefore, there exist
$T^{1}_{\mu},\cdots,T^{l}_{\mu}$ which are $C^{1}$-tori tangent to
$E^{ss}_{g}\oplus E^{uu}_{g}$ such that
$\mathrm{supp}(\mu)=\cup_{i=1}^{l}T^{i}_{\mu}$.

If $\mu$ does not have atomic center disintegration, then $\mu$ is an SRB
measure and this concludes the proof of the theorem.

###  8.2 Reducing the proof of Theorem 8.1 into proving $s$-invariance of the
center conditional measures

In this subsection we will reduce the proof of Theorem 8.1 into proving the
following theorem:

######  Theorem 8.3 (The $s$-invariance of measures with atomic
disintegration).

Let $g\in\mathrm{Sk}^{2}(\mathbb{T}^{2}\times\mathbb{T}^{2})$ be a partially
hyperbolic, center bunched skew product. Suppose that there exists a constant
$\theta\in(0,1)$ such that

$E^{uu}\textrm{ is $\theta$-H\"{o}lder and
}\|Dg|_{E^{ss}}\|^{\theta}<m(Dg|_{E}^{c}).$

Let $\mu$ be an ergodic $u$-Gibbs measure for $g$ and suppose that $\mu$
verifies:

  1. 1.

$\mu$ has atomic center disintegration;

  2. 2.

for $\mu$-almost every $p\in\mathbb{T}^{4}$, and for Lebesgue almost every
point $q$ in $W^{uu}_{loc}(p)$

$E^{-}_{g,q}\neq DH^{u}_{p,q}(p)E^{-}_{g,p}.$

Let $\nu$ be the unique SRB-measure on $\mathbb{T}^{2}$ for the Anosov
diffeomorphism $g_{2}$. Then there exists a set $X$ of full $\nu$-measure such
that for any $p_{2},q_{2}\in X$ in the same stable leaf for $g_{2}$, we have
that

$\mu^{c}_{q_{2}}=(H^{s}_{p_{2},q_{2}})_{*}\mu^{c}_{p_{q}}.$

For the rest of this section we suppose that $g$ and $\mu$ verify the
conditions of Theorem 8.1. Recall that $\mu\in\mathrm{State}^{u}_{\nu}(g)$, by
Corollary 2.30, the measure $\mu$ has $u$-invariant center conditional
measures (see (14) for the definition of $u$-invariant). By Theorem 8.3, the
measure $\mu$ also has $s$-invariant center conditional measures.

Observe that for $g_{2}$, after fixing some small $\varepsilon>0$, for any
$p_{2}\in\mathbb{T}^{2}$ there exists a neighborhood $U$ of $p_{2}$ such that
the map $[.,.]:W^{ss}_{g_{2},\varepsilon}(p_{2})\times
W^{uu}_{g_{2},\varepsilon}(p_{2})\to U$ defined by
$[x^{s},y^{u}]=W^{ss}_{g_{2},\varepsilon}(x^{s})\cap
W^{uu}_{g_{2},\varepsilon}(y^{u})$ is a homeomorphism. For a probability
measure $\hat{\nu}$ on $\mathbb{T}^{2}$ we say that it has local product
structure if for any $p_{2}\in\mathrm{supp}(\hat{\nu})$, using the
homeomorphism $[.,.]$ as above, the measure $\hat{\nu}$ can be written as
$\rho\hat{\nu}^{s}\times\hat{\nu}^{u}$, where $\rho$ is a positive measurable
function.

Since the measure $\nu$ is the unique SRB measure of $g_{2}$, in particular,
it is an equilibrium state for the logarithm of the unstable jacobian of
$g_{2}$, it has local product structure (see [14]). Since the unstable
foliation of $g_{2}$ is minimal (i.e. every unstable manifold is dense), then
the support of $\nu$ is $\mathbb{T}^{2}$. The proof of the following
proposition can be found in [5]. It is a type of Hopf argument, and is a
consequence of the local product structure of $\nu$, the $su$-invariance of
the center conditional measures of $\mu$, and that the support of $\nu$ is the
entire torus.

######  Lemma 8.4.

Let $\mu$ be a measure as above. Then there exists a disintegration
$\\{\overline{\mu}^{c}_{p_{2}}\\}_{p_{2}\in\mathbb{T}^{2}}$ of $\mu$ with the
following properties:

  1. 1.

the measures $\overline{\mu}^{c}_{p_{2}}$ coincides with the measures
$\mu^{c}_{p_{2}}$ for $\nu$-almost every $p_{2}$;

  2. 2.

the map $p_{2}\mapsto\overline{\mu}^{c}_{p_{2}}$ is continuous;

  3. 3.

for any $p_{2}\in\mathbb{T}^{2}$, and $q_{2}\in W^{uu}_{g_{2}}(p_{2})$ we have
$\overline{\mu}^{c}_{q_{2}}=(H^{u}_{p_{2},q_{2}})_{*}\overline{\mu}^{c}_{p_{2}}$;

  4. 4.

for any $p_{2}\in\mathbb{T}^{2}$, and $q_{2}\in W^{ss}_{g_{2}}(p_{2})$ we have
$\overline{\mu}^{c}_{q_{2}}=(H^{s}_{p_{2},q_{2}})_{*}\overline{\mu}^{c}_{p_{2}}$.

By an abuse of notation, we will denote the disintegration
$\\{\overline{\mu}^{c}_{p_{2}}\\}_{p_{2}\in\mathbb{T}^{2}}$ by
$\\{\mu^{c}_{p_{2}}\\}_{p_{2}\in\mathbb{T}^{2}}$. We will also need the
following lemma:

######  Lemma 8.5.

There exists $k\in\mathbb{N}$ such that for every $p_{2}\in\mathbb{T}^{2}$ the
measure $\mu^{c}_{p_{2}}$ has $k$-atoms.

###### Proof.

We already know that the measure $\mu^{c}_{p_{2}}$ is atomic for every
$p_{2}\in\mathbb{T}^{2}$. For each $n\in\mathbb{N}$ consider the set
$B_{n}:=\\{p\in\mathbb{T}^{4}:\mu^{c}_{\pi_{2}(p)}(\\{p\\})>\frac{1}{n}\\}$.
It is easy to see that $B_{n}$ is a $g$-invariant set for each
$n\in\mathbb{N}$. For $n$ sufficiently large $\mu(B_{n})>0$, and by ergodicity
$\mu(B_{n})=1$. Hence, for $n$ large enough, every atom of $\mu^{c}_{p_{2}}$
has measure larger than $\frac{1}{n}$, for any $p_{2}\in\mathbb{T}^{2}$, and
therefore there are finitely many atoms. Let $\mathcal{A}_{p_{2}}$ be the set
of atoms of $\mu^{c}_{p_{2}}$. The $su$-invariance implies that for any
$\\#\mathcal{A}_{p_{2}}=\\#\mathcal{A}_{q_{2}}$, for any
$p_{2},q_{2}\in\mathbb{T}^{2}$. Hence, there exists $k\in\mathbb{N}$ such that
$\mu^{c}_{p_{2}}$ has exactly $k$-atoms, for every $p_{2}\in\mathbb{T}^{2}$. ∎

###### Proof of Theorem 8.1.

Let $k\in\mathbb{N}$ be as in lemma 8.5 and fix $p_{2}\in\mathbb{T}^{2}$. Let
$\mathcal{A}_{p_{2}}:\\{x_{1},\cdots,x_{k}\\}$ be the set of atoms of
$\mu^{c}_{p_{2}}$. The $su$-invariance of
$\\{\mu^{c}_{p_{2}}\\}_{p_{2}\in\mathbb{T}^{2}}$ implies that for any
$x_{i}\in\mathcal{A}_{p_{2}}$, the endpoint of any $su$-path starting in
$x_{i}$ and ending in $W^{c}(x_{i})$ also belongs to $\mathcal{A}_{p_{2}}$. In
particular, for each $x_{i}\in\mathcal{A}_{p_{2}}$, its accessibility class
$AC(x_{i})$ is trivial (see section 2 for the definition of trivial
accessibility class).

There exists $l\in\mathbb{N}$ such that the union of the accessibility classes
of the points $x_{i}$ can be partitioned into $l$ disjoint accessibility
classes, that is,

$\displaystyle\bigcup_{i=1}^{k}AC(x_{i})=T^{1}_{\mu}\sqcup\cdots\sqcup
T^{l}_{\mu},$

where each $T^{i}_{\mu}$ is an accessibility class.

We will prove that each $T^{i}_{\mu}$ is a two dimensional torus tangent to
$E^{ss}_{g}\oplus E^{uu}_{g}$. Since
$\mathrm{supp}(\mu)\subset\bigcup_{i=1}^{k}AC(x_{i})$, this will conclude the
proof of the theorem.

By Theorem 2.14, we know that $T^{i}_{\mu}$ is a $C^{1}$-submanifold of
$\mathbb{T}^{4}$. Furthermore, since the accessibility class $T^{i}_{\mu}$ is
trivial, we obtain that it is a $2$-dimensional $C^{1}$-submanifold immersed
in $\mathbb{T}^{4}$ and by the definition of accessibility class it is tangent
to $E^{ss}_{g}\oplus E^{uu}_{g}$.

######  Claim 8.6.

$T^{i}_{\mu}$ is compact.

###### Proof.

Since the direction $E^{ss}_{g}\oplus E^{uu}_{g}$ is uniformly transverse to
$E^{c}_{g}$, the surface $T^{i}_{\mu}$ is uniformly transverse to the center
foliation. If it was not compact, then it would intersect some center leaf
$W^{c}(p)$ infinitely many times. However, this is a contradiction with the
fact that $T^{i}_{\mu}\cap W^{c}(p)\subset\mathcal{A}_{\pi_{2}(p)}$ which is
finite. ∎

Since the strong stable and strong unstable manifolds of $g$ projects to the
stable and unstable manifolds of $g_{2}$, which are dense in $\mathbb{T}^{2}$
we have that $\pi_{2}(T^{i}_{\mu})=\mathbb{T}^{2}$. Let us see that
$\pi_{2}|_{T^{i}_{\mu}}:T^{i}_{\mu}\to\mathbb{T}^{2}$ is a covering map. The
property that $T^{i}_{\mu}$ is $su$-saturated implies that
$\pi_{2}|_{T^{i}_{\mu}}$ is surjective. Since $T^{i}_{\mu}$ is tangent to
$E^{ss}_{g}\oplus E^{uu}_{g}$, which is uniformly transverse to the fibers
(center direction), then for any point $p_{2}\in\mathbb{T}^{2}$ and small
neighborhood $U$ of $p_{2}$, any connected component of $\pi_{2}^{-1}(U)\cap
T^{i}_{\mu}$ is diffeomorphic to $U$, hence, $\pi_{2}|_{T^{i}_{\mu}}$ is a
covering map, and $T^{i}_{\mu}$ is a cover of $\mathbb{T}^{2}$. The only
possible covers of $\mathbb{T}^{2}$ are homeomorphic to $\mathbb{R}^{2}$,
$S^{1}\times\mathbb{R}$ and $\mathbb{T}^{2}$. Using that $T^{i}_{\mu}$ is
compact, we conclude that $T^{i}_{\mu}$ is actually a two torus. ∎

##  9 The proof of Theorem 8.3: the $s$-invariance of the center conditional
measures

The goal of this section is to prove Theorem 8.3. Recall that for a partially
hyperbolic diffeomorphism $f$ and $\mu$ an $f$-inviariant measure, we defined
in (12) the $\mu$-partial entropy along $\mathcal{F}^{uu}$, which is given by

$h_{\mu}(f,\mathcal{F}^{uu})=-\displaystyle\int\log\mu^{uu}_{p}(f^{-1}\xi^{uu}(p))d\mu(p),$

where $\xi^{uu}$ is any $\mu$-measurable partition subordinated to
$\mathcal{F}^{uu}$.

Let us first see how the proof of Theorem 8.3 is reduced into proving the
following theorem:

######  Theorem 9.1.

Let $g\in\mathrm{Sk}^{2}(\mathbb{T}^{2}\times\mathbb{T}^{2})$ be a partially
hyperbolic, center bunched skew product. Suppose that there exists a constant
$\theta\in(0,1)$ such that

$E^{uu}_{g}\textrm{ is $\theta$-H\"{o}lder and
}\|Dg|_{E^{ss}}\|^{\theta}<m(Dg|_{E^{c}}).$ (32)

Suppose that $\mu$ is an ergodic $u$-Gibbs measure that verifies:

  1. 1.

$\mu$ has both a positive and a negative Lyapunov exponent along the center;

  2. 2.

$\mu$ has atomic center disintegration;

  3. 3.

for $\mu$-almost every $p\in\mathbb{T}^{4}$, and for Lebesgue almost every
point $q$ in $W^{uu}_{loc}(p)$

$E^{-}_{g,q}\neq DH^{u}_{p,q}(p)E^{-}_{g,p}.$

Let $\nu$ be the unique SRB-measure on $\mathbb{T}^{2}$ for the Anosov
diffeomorphism $g_{2}$. Then

$h_{\mu}(g^{-1},\mathcal{F}^{ss})=h_{\nu}(g_{2}).$ (33)

###### Proof of Theorem 8.3 assuming Theorem 9.1.

Let $g$ be a partially hyperbolic skew product and $\mu$ be an ergodic
$u$-Gibbs measure verifying the hypothesis of Theorem 8.3. By Theorem 9.1,
$\mu$ verifies
$h_{\mu}(g^{-1},\mathcal{F}^{ss})=h_{\nu}(g_{2})=h_{\nu}(g_{2}^{-1})$.
Applying the invariance principle (Theorem 2.26), we have that $\mu$ is an
$s$-state projecting on $\nu$. Proposition 2.27 applied to $g^{-1}$ then
implies the conclusion of Theorem 8.3.

∎

######  Remark 9.2.

For the rest of the section we fix

  * •

$g$ a partially hyperbolic skew product and

  * •

$\mu$ an ergodic $u$-Gibbs measure for $g$ that verifies the hypothesis of
Theorem 9.1.

Let

  * •

$\nu$ be the unique SRB measure for $g_{2}$ on $\mathbb{T}^{2}$, such that
$(\pi_{2})_{*}\mu=\nu$, and

  * •

$\lambda^{+}$ and $\lambda^{-}$ be the positive and negative center expoenents
of $\mu$, respectively.

  * •

Fix $\xi^{uu}_{2}$ a $\nu$-measurable partition of $\mathbb{T}^{2}$ which is
subordinated to $\mathcal{F}^{uu}_{2}$ (the unstable foliation of the Anosov
system $g_{2}$).

  * •

Let $\xi^{uu}$ be a $\mu$-measurable partition which is subordinated to
$\mathcal{F}^{uu}$, with the property that
$\pi_{2}(\xi^{uu}(p))=\xi^{uu}_{2}(\pi_{2}(p))$. Furthermore, we may assume
that each element of $\xi^{uu}$ has small diameter.

The proof of Theorem 9.1 follows closely the proof of Theorem $4.8$ in [17],
with some adaptations. Brown-Rodriguez Hertz’s proof is based on the
“exponential drift” arguments that were introduced in [9] and its modified
version from [30]. Since the proof of Theorem $4.8$ in [17] is technical and
to make this work more self contained, we repeat most of the proof here, with
the necessary adaptations. For the proof of some of the lemmas we will refer
the reader to [17].

###  9.1 Sketch of the proof of Theorem 9.1

Let us describe the idea of the proof as well as the main points where our
setting differs from the random product setting.

Suppose that $\mu$ is a $u$-Gibbs measure verifying the hypothesis Theorem
9.1. We fix a compact set $K$ of large measure such that the map
$p\mapsto\mu^{c}_{p}$ is continuous. Since $\mu^{c}_{p}$ is atomic, in this
set, we can fix a constant $\varepsilon>0$ such that any two atoms in a center
leaf have distance greater than $\varepsilon$.

Suppose that the conclusion of the theorem does not hold, then we obtain that
conditional measures along the $2$-dimensional Pesin stable manifolds are not
supported in a single strong stable leaf. In particular, we can find two
points $p,q$ in the same $2$-dimensional stable manifold with
$d(p,q):=\delta<<\varepsilon$ such that they don’t belong to the same strong
stable leaf, $p$ is an atom of $\mu^{c}_{p}$ and $q$ is an atom for
$\mu^{c}_{q}$. The idea of the argument is to find a point $p_{0}\in K$ such
that $\mu^{c}_{p_{0}}$ has two atoms with distance smaller than $\varepsilon$,
which will give a contradiction with the choice of $\varepsilon$ and $K$.

The idea is to find two sequences of times $l_{j}\to+\infty$ and
$\tau_{j}\to+\infty$ such that for each $l_{j}$, we can choose two points
$\overline{p}_{j}\in W^{uu}_{loc}(g^{l_{j}}(p))$ and $\overline{q}_{j}\in
W^{uu}_{loc}(g^{l_{j}}(q))$ that verify the following:

  1. 1.

$\overline{q}_{j}\in W^{cs}(\overline{p}_{j})$;

  2. 2.

let
$\overline{w}_{j}=H^{s}_{\overline{q}_{j},\overline{p}_{j}}(W^{-}_{loc}(\overline{q}_{j}))\cap
W^{+}_{loc}(\overline{p}_{j})$, then
$d(\overline{p}_{j},\overline{w}_{j})\approx d(g^{l_{j}}(p),g^{l_{j}}(q))$;

  3. 3.

$d(g^{\tau_{j}}(\overline{p}_{j}),g^{\tau_{j}}(\overline{q}_{j}))\approx\delta.$

Then we prove that, up to a subsequence, $g^{\tau_{j}}(\overline{p}_{j})$ and
$g^{\tau_{j}}(\overline{q}_{j})$ converge to $p_{0}$ and $q_{0}$ which are
atoms of $\mu^{c}_{p_{0}}$. Item $3$ above implies that
$d(p_{0},q_{0})\approx\delta<<\varepsilon$. Then we show that we can do this
so that $p_{0}$ belongs to the set of points for which the distance of any two
atoms of $\mu^{c}_{p_{0}}$ is greater than $\varepsilon$ and this will give
the contradiction.

Let me remark that the hypothesis that $E^{-}$ is not $DH^{u}$ invariant
appears to obtain item $2$, and item $2$ is used to prove item $3$.

This is the strategy used in the proof of Theorem $4.8$ from [17]. As we
follow this proof, let us mention the two points where more adaptation is
needed in our setting.

  * •

The main difficult in the strategy is to work this construction so that all
the points involved belong to some “good” set. To achieve that, in their
proof, they work with the suspension flow and a reparametrization of this
flow. This reparametrized flow is convenient because it gives precise times
where a certain expansion is observed along the Oseledets unstable direction.
One of the key tools they use to control these returns to the “good” set is a
martingale convergence argument for this reparametrized flow (see Sections
$9.3$ and $9.4$ in [17]), in which they apply the reverse martingale
convergence theorem. To apply this theorem, they need that the reparametrized
flow verifies some measurability condition.

In our setting, we use $DH^{u}$ to adjust the definition of the reparametrized
flow so that the measurability condition is verified. This is done in Section
9.4.2. Let me remark that this is similar to the change of coordinates done in
Section 6.2.

  * •

The other point where some adaptation is needed is to obtain item $2$ above.
Let us briefly explain Brown-Rodriguez Hertz’s proof of item $2$. Let
$\overline{z}_{j}=H^{s}_{\overline{q}_{j},\overline{p}_{j}}(W^{-}_{loc}(\overline{q}_{j}))\cap
H^{u}_{g^{l_{j}}(p),\overline{p}_{j}}(W^{-}_{loc}(g^{l_{j}}(p)))$. Let
$q^{*}:=H^{s}_{q,p}(q)$ and observe that $q^{*}\in W^{-}_{loc}(p)$. If we had
local $su$-integrability, we would have that
$\overline{z}_{j}=H^{u}_{g^{l_{j}}(p),\overline{p}_{j}}(g^{l_{j}}(q^{*}))$,
since $\overline{q}_{j}=H^{u}_{q_{j},\overline{q}_{j}}\circ
H^{s}_{p_{j},q_{j}}(g^{l_{j}}(q^{*}))$. Then, by a geometrical argument, we
could conclude that $d(\overline{p}_{j},\overline{w}_{j})\approx
d(\overline{p}_{j},H^{u}_{g^{l_{j}}(p),\overline{p}_{j}}(g^{l_{j}}(q^{*}))\approx
d(g^{l_{j}}(p),g^{l_{j}}(q^{*})).$ In the random product setting considered in
[17], this local “joint integrability” is verified, since these holonomies
maps are just the identity between fibers.

In our setting, to obtain the estimate in item $2$, we need to use two
ingredients: the unstable foliation is $\theta$-Hölder, and
$\|Dg|_{E^{ss}}\|^{\theta}<m(Dg|_{E^{c}})$. This is where condition (32) comes
in. The estimate needed for item $2$ is obtained in Lemma 9.17.

###  9.2 Pesin Theory and parametrization of invariant manifolds

From now on fix a constant
$0<\varepsilon_{0}\ll\min\\{1,-\lambda^{-},\lambda^{+}\\}$. We will be
interested in obtaining certain estimates for stable and unstable manifolds
along the center. In our setting, these will correspond to curves contained in
horizontal tori.

On $\mathbb{R}^{2}$ consider the the standard basis and for any
$v\in\mathbb{R}^{2}$ write $v=v_{1}+v_{2}$. Consider the metric on
$\mathbb{R}^{2}$ given by $|v|=\max\\{|v_{1}|,|v_{2}|\\}$. For any $l>0$ write
$\mathbb{R}^{2}(l)$ to be the ball of radius $l$ centered at the origin for
this metric. Fix $0<\varepsilon_{1}<\varepsilon_{0}$. There exists a
measurable function $l:\mathbb{T}^{2}\times\mathbb{T}^{2}\to[1,+\infty)$, and
a $\mu$-full measurable set $\Lambda$ such that:

  1. 1.

For each point $p=(p_{1},p_{2})\in\Lambda$, there is a neighborhood
$U_{p}\subset\mathbb{T}^{2}\times\\{p_{2}\\}$ of $p_{1}$, and a diffeomorphism
$\phi_{p}:U_{p}\to\mathbb{R}^{2}(l(p)^{-1})$ with:

    1. (a)

$\phi_{p}(p_{1})=0$;

    2. (b)

$D\phi_{p}(p_{1})E^{-}_{p}=\mathbb{R}\times\\{0\\}$;

    3. (c)

$D\phi_{p}(p_{1})E^{+}_{p}=\\{0\\}\times\mathbb{R}$.

  2. 2.

Let

$\tilde{g}_{p}=\phi_{g(p)}\circ g\circ\phi^{-1}_{p}\textrm{ and
}\tilde{g}^{-1}_{p}=\phi_{g^{-1}(p)}\circ g^{-1}\circ\phi^{-1}_{p}.$

On the domain of definition of $\tilde{g}_{p}$, we have:

    1. (a)

$\tilde{g}_{p}(0)=0$;

    2. (b)

$D\tilde{g}_{p}(0)=\begin{pmatrix}\beta^{-}&0\\\ 0&\beta^{+}\end{pmatrix}$,
where
$\beta^{-}\in(e^{\lambda^{-}-\varepsilon_{1}},e^{\lambda^{-}+\varepsilon_{1}})$,
and
$\beta^{+}\in(e^{\lambda^{+}-\varepsilon_{1}},e^{\lambda^{+}+\varepsilon_{1}})$.

    3. (c)

Writing $\mathrm{Lip}(.)$ the Lipschitz constant of a function in its domain
of definition, we have
$\mathrm{Lip}(\tilde{g}_{p}-D\tilde{g}_{p}(0))<\varepsilon_{1}$;

    4. (d)

$\mathrm{Lip}(D\tilde{g}_{p}(0))<l(p)$;

    5. (e)

similar property holds for $\tilde{g}^{-1}_{p}$.

  3. 3.

There exists an uniform $k_{0}$ such that
$k_{0}^{-1}\leq\mathrm{Lip}(\phi_{p})\leq l(p)$;

  4. 4.

$l(g^{n}(p))<e^{|n|\varepsilon_{1}}l(p)$, for any $n\in\mathbb{Z}$.

The diffeomorphisms $\phi$ above are called Lyapunov charts. Its construction
can be found for instance in the appendix of [41].

We will also use a more quantified statement of the Pesin’s stable manifold
theorem. Let $\mathbb{R}^{-}:=\mathbb{R}\times\\{0\\}$ and
$\mathbb{R}^{+}:=\\{0\\}\times\mathbb{R}$.

######  Theorem 9.3 (Local stable manifold theorem).

For each $p\in\Lambda$, there exists a $C^{2}$-function
$\varphi^{-}_{p}:\mathbb{R}^{-}(l(p)^{-1})\to\mathbb{R}^{+}(l(p)^{-1})$ such
that:

  1. 1.

$\varphi^{-}_{p}(0)=0$;

  2. 2.

$D\varphi^{-}_{p}(0)=0$;

  3. 3.

$\|D\varphi^{-}_{p}\|<\frac{1}{3}$;

  4. 4.

$\tilde{g}_{p}(\mathrm{graph}(\varphi^{-}_{p}))\subset\mathrm{graph}(\varphi^{-}_{g(p)})\subset\mathbb{R}^{2}(l(g(p))^{-1})$;

  5. 5.

setting $W^{-}_{loc}(p):=\phi^{-1}_{p}(\mathrm{graph}(\varphi^{-}_{p}))$, we
have that

    1. (a)

$g(W^{-}_{loc}(p))\subset W^{-}_{loc}(g(p))$;

    2. (b)

for any $z,y$ in $W^{-}_{loc}(p)$, and $n\geq 0$, we have

$d(g^{n}(z),g^{n}(y))\leq l(p)k_{0}e^{(\lambda^{-}+2\varepsilon_{1})n}d(z,y).$

Similarly, there exists a $C^{2}$-function $\varphi^{+}_{p}$ which will define
the local unstable manifold.

We may define the global stable manifold of $p$ by $W^{-}(p):=\cup_{n\geq
0}g^{-n}(W^{-}_{loc}(g^{n}(p)))$.

In our setting, for $\mu$-almost every point $p$, the stable manifold
$W^{-}(p)$ is a one dimensional curve, and it can be parametrized by
$\mathbb{R}$. We remark that this curve can also be obtained by intersecting a
stable Pesin manifold, which in our setting has dimension two, with a center
manifold.

For these one dimensional stable manifolds, it is convenient to use some
special parametrization that conjugates the dynamics $g^{n}|_{W^{-}(p)}$ with
the linear dynamics $Dg^{n}(p)|_{E^{-}_{p}}$.

######  Proposition 9.4.

For $\mu$-almost every $(p_{1},p_{2})$, and for any $(q_{1},p_{2})\in
W^{-}(p)$, there exists a $C^{2}$-diffeomorphism

$h^{-}_{(q_{1},p_{2})}:W^{-}(p)\to T_{q_{1}}W^{-}(p),$

such that

  1. 1.

Restricted to $W^{-}(p)$ we have

$Dg(q_{1},p_{2})\circ h^{-}_{(q_{1},p_{2})}=h^{-}_{g(q_{1},p_{2})}\circ g;$
(34)

  2. 2.

$h^{-}_{(q_{1},p_{2})}((q_{1},p_{2}))=0$ and
$Dh^{-}_{(q_{1},p_{2})}((q_{1},p_{2}))=Id$;

The proof of Proposition 9.4 follows from the construction of the
parametrizations that appeared in [37] Section $3.1$ (see also Proposition
$6.5$ in [17]).

For each $r>0$ and $p$ a point that verifies Proposition 9.4, we define

$W^{-}_{r}(p):=\left(h^{-}_{p}\right)^{-1}(\\{v\in E^{-}_{p}:\|v\|<r\\}).$
(35)

One obtains similarly functions $h^{+}_{.}$ and define
$W^{+}_{r}(p):=(h_{p})^{-1}(\\{v\in E^{+}_{p}:\|v\|<r\\})$.

We fix two $\mu$-measurable unitary vector fields $p\mapsto v^{-}_{p}$ and
$p\mapsto v^{+}_{p}$ such that

  * •

$v^{*}_{p}\in E^{*}_{p}$, for $*=-,+$;

  * •

for each $p$ and $q\in\xi^{uu}(p)$ we have that
$v^{+}_{q}=\frac{DH_{p,q}(p).v^{+}_{p}}{\|DH_{p,q}(p).v^{+}_{p}\|}$;

Using these measurable vector fields, we parametrize the stable and unstable
manifolds by

$\mathcal{I}_{p}^{-}:t\mapsto(h^{-}_{p})^{-1}(tv^{-}_{p})\textrm{ and
}\mathcal{I}_{p}^{+}:t\mapsto(h^{+}_{p})^{-1}(tv^{+}_{p}).$ (36)

It is convenient to consider another norm, called Lyapunov norm, where we can
see contraction, or expansion, after one iterate. Let $X$ be a set of full
$\mu$-measure where the Lyapunov exponents are well defined. For each $p\in
X$, and $v\in E^{\sigma}_{p}$ consider the two-sided Lyapunov norm

$\|v\|^{\sigma}_{\varepsilon_{0},\pm,p}:=\left(\displaystyle\sum_{j\in\mathbb{Z}}\|Dg^{j}(p)v\|^{2}e^{-2\lambda^{\sigma}j-2\varepsilon_{0}|j|}\right)^{\frac{1}{2}},\textrm{
where $\sigma=\\{-,+\\}$}$ (37)

and for $v\in E^{+}_{p}$, consider the one-sided Lyapunov norm

$\|v\|_{\varepsilon_{0},-,p}:=\left(\displaystyle\sum_{j\leq
0}\|Dg^{j}(p)v\|^{2}e^{-2\lambda^{+}j-2\varepsilon_{0}|j|}\right)^{\frac{1}{2}}$
(38)

For these norms, we have the following estimates (see [16] for the estimate on
the two-sided norm).

######  Lemma 9.5.

For $p\in X$, and $v\in E^{c}_{p}$, we have that

$\begin{array}[]{rclcl}e^{k\lambda^{-}-|k|\varepsilon_{0}}\|v\|^{-}_{\varepsilon_{0},\pm,p}&\leq&\|Dg^{k}(p)v\|^{-}_{\varepsilon_{0},\pm,g^{k}(p)}&\leq&e^{k\lambda^{-}+|k|\varepsilon_{0}}\|v\|^{-}_{\varepsilon_{0},\pm,p}\\\
e^{k\lambda^{+}-|k|\varepsilon_{0}}\|v\|^{+}_{\varepsilon_{0},\pm,p}&\leq&\|Dg^{k}(p)v\|^{+}_{\varepsilon_{0},\pm,g^{k}(p)}&\leq&e^{k\lambda^{+}+|k|\varepsilon_{0}}\|v\|^{+}_{\varepsilon_{0},\pm,p}.\end{array}$

If $v\in E^{+}_{p}$ then

$e^{k\lambda^{+}-k\varepsilon_{0}}\|v\|_{\varepsilon,-,p}\leq\|Dg^{k}(p)v\|_{\varepsilon_{0},-,g^{k}(p)}.$

The following lemma is a classical lemma in Pesin theory on the control of
expansion/contraction and angle between expanding and contracting directions.

######  Lemma 9.6.

There exists a measurable function $L:X\to\mathbb{R}$, such that for any $p\in
X$ and $n\in\mathbb{Z}$:

  1. 1.

For $v\in E^{-}_{p}$,

$\frac{1}{L(p)}e^{n\lambda^{-}-\frac{|n|}{2}\varepsilon_{0}}\|v\|\leq\|Dg^{n}(p)v\|\leq
L(p)e^{n\lambda^{-}+\frac{|n|}{2}\varepsilon_{0}}\|v\|.$

  2. 2.

For $v\in E^{+}_{p}$,

$\frac{1}{L(p)}e^{n\lambda^{+}-\frac{|n|}{2}\varepsilon_{0}}\|v\|\leq\|Dg^{n}(p)v\|\leq
L(p)e^{n\lambda^{+}+\frac{|n|}{2}\varepsilon_{0}}\|v\|.$

  3. 3.

$\measuredangle(E^{+}_{g^{n}(p)},E^{-}_{g^{n}(p)})\geq\frac{1}{L(p)}e^{-|n|\varepsilon_{0}}$.
Furthermore, $L(g^{n}(p))\leq L(p)e^{|n|\varepsilon_{0}}$.

###  9.3 Angles and some estimates

Recall that in our setting there is a set of full $\mu$-measure such that for
any two points $p$ and $q$ in this set with $q\in W^{uu}(p)$ we have that

$DH^{u}_{p,q}(p)E^{-}_{p}\neq E^{-}_{q}.$ (39)

Recall also that $X$ is the set of points of full $\mu$-measure with well
defined Lyapunov exponents.

Given $\gamma_{1}>0$ consider $\Lambda_{1}$ to be the set of points $p$ such
that

$\measuredangle(E^{-}_{p},E^{+}_{p})>\gamma_{1},$ (40)

where $\measuredangle(E^{-}_{p},E^{+}_{p})$ is the angle for the natural
riemannian metric of $\mathbb{T}^{4}$ between the subspaces $E^{-}_{p}$ and
$E^{+}_{p}$. Observe that we can make the $\mu$-measure of $\Lambda_{1}$
arbitrarily close to $1$, by taking $\gamma_{1}$ small.

For $\gamma_{2}\in(0,\frac{\gamma_{1}}{2})$ and $p\in\Lambda_{1}$, we define
$\mathscr{A}_{\gamma_{2}}(p)$ to be the set of points $q\in\xi^{uu}(p)$ such
that

  * •

$q\in X$;

  * •

$\measuredangle(DH^{u}_{p,q}(p)E^{-}_{p},E^{-}_{q})>\gamma_{2}$;

  * •

$\measuredangle(E^{+}_{q},E^{-}_{q})>\gamma_{2}.$

Recall that for $p$ in a set of full $\mu$-measure, we defined $\mu^{uu}_{p}$
as the conditional measure on $\xi^{uu}(p)$ given by the disintegration of
$\mu$ on the partition $\xi^{uu}$.

For each $\gamma_{1},\gamma_{2}$ as above and $a\in(0,1)$, define

$\mathcal{A}_{\gamma_{1},\gamma_{2},a}:=\\{p\in\Lambda_{1}:\mu^{uu}_{p}(\mathscr{A}_{\gamma_{2}}(p))>1-a.\\}.$
(41)

######  Remark 9.7.

By (39), for any two numbers $a,c\in(0,1)$ there exist $\gamma_{1}>0$ and
$\gamma_{2}\in(0,\frac{\gamma_{1}}{2})$ sufficiently small such that
$\mu(\mathcal{A}_{\gamma_{1},\gamma_{2},a})>1-c$.

By Lusin’s theorem, there is a compact set $\Lambda_{2}\subset\mathbb{T}^{4}$
with measure arbitrarily close to $1$ such that the parametrized stable and
unstable manifolds $W^{-}_{r}(p)$ and $W^{+}_{r}(p)$ vary continuously in the
$C^{1}$-topology, for $p\in\Lambda_{2}$, on the space of embeddings
$C^{1}([-r,r],\mathbb{T}^{2})$ for any $r\in(0,1)$.

Given $\theta^{\prime}\in(0,\pi)$ and a one-dimensional space $E$ in
$\mathbb{R}^{2}$, let $\mathscr{C}_{\theta^{\prime}}(E)$ be the cone centered
in $E$ with angle $\theta^{\prime}$. In what follows, we will consider $\exp$
to be the exponential map on $\mathbb{T}^{2}$, and we identify every center
manifold with the two torus $\mathbb{T}^{2}$.

######  Lemma 9.8.

Given $\theta^{\prime}\in(0,\pi)$, there exist $\hat{r}_{0},\hat{r}_{1}>0$
such that for any two points $p=(p_{1},p_{2})$ and $q=(q_{1},q_{2})$ both
belonging to $\Lambda_{2}$, such that $d(p,q)<\hat{r}_{0}$ and $q_{2}\in
W_{g_{2}}^{ss}(p_{2})$, we have:

  1. 1.

$\exp^{-1}_{p_{1}}(W^{*}_{\hat{r}_{1}}(p))\subset\mathscr{C}_{\theta^{\prime}}(E^{*}_{p})$,
for $*=-$ or $+$;

  2. 2.

$\exp^{-1}_{p_{1}}(H^{s}_{q_{2},p_{2}}(W^{-}_{\hat{r}_{1}}(q)))\subset\mathscr{C}_{\theta^{\prime}}(E^{-}_{p})+\exp^{-1}_{p_{1}}(q_{1})$.

###### Proof.

Recall that on $\Lambda_{2}$, for $r\in(0,1)$, the map $\Lambda_{2}\ni
p\mapsto W^{*}_{r}(p)$ varies continuously on the space of embeddings
$C^{1}([-r,r],\mathbb{T}^{2})$. Hence, we may fix $\hat{r}_{1}$ sufficiently
small such that for each $p\in\Lambda_{2}$ we have that
$\exp^{-1}_{p_{1}}(W^{*}_{\hat{r}_{1}}(p))\subset\mathscr{C}_{\frac{\theta}{4}}(E^{*}_{p})$,
for $*=-$ or $+$.

Using that $p\mapsto E^{-}_{p}$ varies continuously on $\Lambda_{2}$, one may
take $\hat{r}_{0}$ sufficiently small so that if $d(p,q)<\hat{r}_{0}$ then
$E^{-}_{q}\subset\mathscr{C}_{\frac{\theta}{4}}(E^{-}_{p})$. Recall that if
$p_{2}$ and $q_{2}$ are in the same strong stable manifold of size $1$ for
$g_{2}$, then $d_{C^{1}}(H^{s}_{p_{2},q_{2}},Id)\leq Cd(p_{2},q_{2})$, for
some constant $C\geq 1$. If $\hat{r}_{0}$ is sufficiently small we also have
that
$\exp^{-1}_{p_{1}}(H^{s}_{q_{2},p_{2}}(W^{-}_{\hat{r}_{1}}(q)))\subset\mathscr{C}_{\frac{\theta}{2}}(DH^{s}_{q_{2},p_{2}}(q)E^{-}_{p})$,
and
$DH^{s}_{q_{2},p_{2}}(q)E^{-}_{q}\subset\mathscr{C}_{\frac{\theta}{2}}(E^{-}_{p})$.
This implies the second item of the lemma. ∎

For each $l_{0}$, we may consider the set $\Lambda_{3}\subset\Lambda_{2}$ of
points having the value $l(p)$ bounded above by $l_{0}$, where
$l:\mathbb{T}^{2}\times\mathbb{T}^{2}\to[1,+\infty)$ is the function defined
in Section 9.2. We may also fix $\tilde{r}_{0}>0$ and $\tilde{r}_{1}>0$ small
enough such that for each $p\in\Lambda_{3}$, we have

  1. (a)

$W^{-}_{\tilde{r}_{1}}(p)\subset W^{-}_{loc}(p)$, where $W^{-}_{loc}(p)$ is
the local stable manifold defined in Theorem 9.3;

  2. (b)

for $q=(q_{1},q_{2})\in\Lambda_{2}$ such that $q_{2}\in
W^{ss}_{g_{2}}(p_{2})$, if $d(p,q)<\tilde{r}_{0}$ then
$\phi_{p}^{-1}(H^{s}_{q_{2},p_{2}}(W^{-}_{\tilde{r_{1}}}(q))$ is contained in
the graph of a $1$-Lipschitz function
$G:D\subset\mathbb{R}^{-}\to\mathbb{R}^{+}$, where
$D\subset\mathbb{R}^{-}(l_{0}^{-1})$.

The second point above follows from combining the estimates from Theorem 9.3
and Lemma 9.8.

Observe that by taking $l_{0}$ large, the set $\Lambda_{3}$ has $\mu$-measure
arbitrarily close to $\mu(\Lambda_{2}).$

######  Lemma 9.9.

For every $\gamma_{1}>0$, $\gamma_{2}\in(0,\frac{\gamma_{1}}{2})$ and
$\Lambda_{3}\subset\Lambda_{2}\subset\Lambda_{1}$ as above, there exist a
measurable set $\Lambda^{\prime}\subset\Lambda_{3}$, with
$\mu(\Lambda^{\prime})$ arbitrarily close to $\mu(\Lambda_{3})$, constants
$r_{0}\in(0,\tilde{r}_{0})$, $r_{1}\in(0,\tilde{r}_{1})$, and
$C_{1},C_{2},C_{3}>1$, with the following properties: For each
$p\in\Lambda^{\prime}$ we have

  1. 1.

$\frac{1}{C_{1}}d(p,w)\leq\|h^{*}_{p}(w)\|\leq C_{1}d(p,w)$, for every $w\in
W^{*}_{r_{1}}(p)$, with $*=-$ or $+$, where $h^{*}$ is given by Proposition
9.4;

Let $p=(p_{1},p_{2})\in\Lambda^{\prime}$,
$p^{\prime}=(p_{1}^{\prime},p_{2}^{\prime})\in\mathscr{A}_{\gamma_{2}}(p)\cap\Lambda^{\prime}$
and $q=(q_{1},q_{2})\in\Lambda^{\prime}$ such that $q\in
W^{cs}_{r_{0}}(p^{\prime})$ and $d(p^{\prime},q)<r_{0}$. Then,

  1. 2.

$W^{+}_{r_{1}}(p^{\prime})\cap H^{s}_{q_{2},p_{2}^{\prime}}(W^{-}_{r_{1}}(q))$
is a single point $w^{\prime}$, furthermore this intersection is transverse
with angles uniformly bounded away from zero inside $T_{x}\mathbb{T}^{2}$;

  2. 3.

$H^{u}_{p_{2},p_{2}^{\prime}}(W^{-}_{r_{1}}(p))\cap
H^{s}_{q_{2},p_{2}^{\prime}}(W^{-}_{r_{1}}(q))$ is a unique point
$z^{\prime}$, this intersection is transverse with angle uniformly bounded
from below.

  3. 4.

For $z^{\prime}$ and $w^{\prime}$ as above,

$\frac{1}{C_{2}}d(p^{\prime},z^{\prime})\leq d(p^{\prime},w^{\prime})\leq
C_{2}d(p^{\prime},z^{\prime});$

  4. 5.

$\displaystyle\frac{1}{C_{3}}\leq\frac{\|Dg^{n}(w^{\prime})|_{T_{w^{\prime}}W^{+}_{r_{1}}(p^{\prime})}\|}{\|Dg^{n}(q)|_{E^{+}_{q}}\|}\leq
C_{3}$, for every $n\geq 0$.

  5. 6.

For any $x\in\Lambda^{\prime}$ and $y\in W^{+}_{r_{1}}(x)$ we have that

$\displaystyle\frac{1}{C_{3}}\leq\frac{\|Dg^{-n}(y)|_{T_{y}W^{+}_{r_{1}}(x)}\|}{\|Dg^{-n}(x)|_{E^{+}_{x}}\|}\leq
C_{3},\textrm{ for every $n\geq 0$.}$

###### Proof.

Items $1-3$ follow from $C^{1}$-topology, Luzin’s theorem and using that in
our setting there exists a constant $C>1$ such that $x_{2}\in
W^{ss}_{g_{2}}(y_{2})$ with $d(x_{2},y_{2})<1$, we have
$d(H^{s}_{x_{2},y_{2}},Id)<C.d(x_{2},y_{2})$ (similar estimate holds for
unstable holonomies). One can also conclude that for any $r_{1}>0$ small, if
$r_{0}>0$ is sufficiently small the conclusions hold.

To prove item $4$, by items $2$ and $3$ above, the angles of the intersections
$W^{+}_{r_{1}}(p^{\prime})\cap H^{s}_{q_{2},p_{2}^{\prime}}(W^{-}_{r_{1}}(q))$
and $H^{u}_{p_{2},p_{2}^{\prime}}(W^{-}_{r_{1}}(p))\cap
H^{s}_{q_{2},p_{2}^{\prime}}(W^{-}_{\tilde{r}_{1}}(q))$ are uniformly bounded
away from zero (depending on $\gamma_{1},\gamma_{2}$ that are fixed).

Fix $\theta^{\prime}>0$ small. We may suppose that $r_{1}<\hat{r}_{1}$ and
$r_{0}<\hat{r}_{0}$ are sufficiently small so that items $1-3$ remain valid,
where $\hat{r}_{0},\hat{r}_{1}$ are the constants given by Lemma 9.8. From the
conclusion of Lemma 9.8, we obtain that for any $x_{1}\in
W^{+}_{r_{1}}(p^{\prime})$, $x_{2}\in
H^{s}_{q_{2},p_{2}^{\prime}}(W^{-}_{r_{1}}(q))$, and $x_{3}\in
H^{u}_{p_{2},p_{2}^{\prime}}(W^{+}_{r_{1}}(p))$, we have that the angles
between $T_{x_{1}}W^{+}_{r_{1}}(p^{\prime})$,
$T_{x_{2}}H^{s}_{q_{2},p_{2}^{\prime}}(W^{-}_{r_{1}}(q))$, and
$T_{x_{3}}H^{u}_{p_{2},p_{2}^{\prime}}(W^{+}_{r_{1}}(p))$ are uniformly
bounded from below. The estimate in item $4$ then follows from the law of
sines (see Figura 2).

Figure 2: Comparing $d(p^{\prime},w^{\prime})$ and $d(p^{\prime},z^{\prime})$.

Since in $\Lambda_{3}$ the function $l(.)$ is bounded by $l_{0}$, by the
properties given by the Lyapunov charts, we obtain that there exists a
constant $C(l_{0})$ such that for any $n\geq 0$,

$\displaystyle d(T_{g^{-n}(y)}W^{+}_{r_{1}}(g^{-n}(x)),E^{+}_{g^{-n}(x)})\leq
C(l_{0})\left(\frac{e^{\lambda^{-}+\varepsilon_{1}}}{e^{\lambda^{+}-\varepsilon_{1}}}\right)^{n}e^{2\varepsilon_{1}n}.$

We also have that
$d(g^{-n}(x),g^{-n}(y))<l_{0}k_{0}e^{(-\lambda^{+}+2\varepsilon_{1})n}d(x,y)$
for $n\geq 0$. In what follows denote $T_{g^{-j}(y)}W^{+}(g^{-j}(x))$ by
$E_{y_{-j}}$, $x_{-j}=g^{-j}(x)$ and $y_{-j}=g^{-j}(y)$, for any
$j\in\mathbb{N}$. Observe that

$\|Dg^{-1}(y_{-j})|_{E_{y_{-j}}}\|\leq\|g\|_{C^{2}}\|Dg^{-1}(x_{-j})|_{E^{+}_{x_{-j}}}\|\max\\{d(x_{-j},y_{-j}),d(E_{y_{-j}},E^{+}_{x_{-j}})\\}.$

Since

$\displaystyle\frac{\|Dg^{-n}(y)|_{E_{y_{0}}}\|}{\|Dg^{-n}(x)|_{E^{+}_{x}}\|}=\prod_{j=0}^{n-1}\frac{\|Dg^{-1}(y_{-j})|_{E_{y_{-j}}}\|}{\|Dg^{-1}(x_{-j})|_{E^{+}_{x_{-j}}}\|},$

the result then follows combining the estimates above.

The proof of item $5$ is similar to the proof of item $6$. One uses the
information that the future orbit of the points $w^{\prime}$ and $q$ converge
exponentially and that the respective tangent directions considered also
converge uniformly exponentially fast on $\Lambda_{3}$.

∎

###  9.4 Reparametrized suspension flow, stopping times and the Martingale
convergence argument

####  9.4.1 The suspension flow

It will be convenient for us to work with the suspension flow associated with
$g$ and a reparametrization of it. Let us recall the definition of the
suspension flow.

Consider the $5$-dimensional manifold
$\tilde{M}=\mathbb{T}^{4}\times\mathbb{R}$. On $\tilde{M}$ consider the
following equivalence relation

$(p,l)\sim(g(p),l-1).$

Let $M=\tilde{M}/\sim$ be the quotient manifold, and consider the flow
$\Phi_{t}:M\to M$ defined by $\Phi_{t}([p,l])=[p,l+t]$, where $[p,l]$ denotes
the equivalence class of the point $(p,l)\in\tilde{M}$. For $\zeta=[p,l]\in
M$, we consider the center fiber $\mathbb{T}^{2}_{\zeta}$ which is given by
the projection of
$\mathbb{T}^{2}\times\\{p_{2}\\}\times\\{l\\}\subset\tilde{M}$ into $M$. The
fiber $\mathbb{T}^{2}_{\zeta}$ is naturally identified with $\mathbb{T}^{2}$.
We will use the coordinates on $M$ induced by $\mathbb{T}^{4}\times[0,1)$.

Consider the measure on $\tilde{M}$ defined by
$\tilde{\omega}:=\mu\times\mathrm{Leb}_{\mathbb{R}}$, where $\mu$ is the
measure on $\mathbb{T}^{4}$ and $\mathrm{Leb}_{\mathbb{R}}$ is the usual
Lebesgue measure on $\mathbb{R}$. This measure projects to a probability
measure $\omega$ on $M$, which in the coordinates $\mathbb{T}^{4}\times[0,1)$
can be written as $d\omega(p,l)=d\mu(p)dl$. Observe that this measure is
invariant by the flow $\Phi_{t}$. Recall that $X$ is the set of full
$\mu$-measure on $\mathbb{T}^{4}$ where the Lyapunov exponents are well
defined. Let $Y$ be the projection on $M$ of the set $X\times\mathbb{R}$
defined on $\tilde{M}$. This is a set of full $\omega$-measure and for each
$\zeta=[p,l]\in Y$, we may define the Oseledets splitting of the center
direction $T_{\zeta}\mathbb{T}^{2}_{\zeta}=E^{-}_{\zeta}\oplus E^{+}_{\zeta}$,
where $E^{*}_{\zeta}=D\Phi_{t}(\zeta).E^{\sigma}_{p}$, for $*=-,+$.

We can naturally extend to $Y$ the vector fields $p\mapsto v_{p}^{*}$ (defined
in Section 9.2) and the parametrizations defined in (36). We can also extend
to $Y$ the Lyapunov norms defined in (37) in the following way. Let
$\zeta=[p,l]\in Y$, then for any $v\in E^{\sigma}_{\zeta}$ we define

$\|v\|^{*}_{\varepsilon_{0},\pm,\zeta}:=\displaystyle\left(\|v\|^{*}_{\varepsilon_{0},\pm,p}\right)^{1-l}\left(\|Dg(p)v\|_{\varepsilon_{0},\pm,g(p)}^{*}\right)^{l},\textrm{
for $*=-,+$.}$ (42)

We define in a similar way the one-sided norm
$\|v\|_{\varepsilon_{0},-,\zeta}$, for $v\in E^{+}_{\zeta}$. This norm allows
us to have expansion or contraction varying continuously with the time. In
particular, from the estimates of Lemma 9.5, we obtain:

######  Lemma 9.10.

For $\zeta=[p,l]\in Y$, for $v\in T_{\zeta}\mathbb{T}^{2}_{\zeta}$, and for
any $t\in\mathbb{R}$ we have that

$\begin{array}[]{rclcl}e^{t\lambda^{-}-|t|\varepsilon_{0}}\|v\|^{-}_{\varepsilon_{0},\pm,\zeta}&\leq&\|D\Phi_{t}(\zeta)v\|^{-}_{\varepsilon_{0},\pm,\Phi_{t}(\zeta)}&\leq&e^{t\lambda^{-}+|t|\varepsilon_{0}}\|v\|^{-}_{\varepsilon_{0},\pm,\zeta}\\\
e^{t\lambda^{+}-|t|\varepsilon_{0}}\|v\|^{+}_{\varepsilon_{0},\pm,\zeta}&\leq&\|D\Phi_{t}(\zeta)v\|^{+}_{\varepsilon_{0},\pm,\Phi_{t}(\zeta)}&\leq&e^{t\lambda^{+}+|t|\varepsilon_{0}}\|v\|^{+}_{\varepsilon_{0},\pm,\zeta}.\end{array}$

If $v\in E^{+}_{\zeta}$ then

$e^{t\lambda^{+}-|t|\varepsilon_{0}}\|v\|_{\varepsilon,-,\zeta}\leq\|D\Phi_{t}(\zeta)v\|_{\varepsilon_{0},-,\Phi_{t}(\zeta)}.$

Recall that $L(.)$ is the function from Lemma 9.6. The proof of the following
lemma can be obtained by a simple adaptation of the proof of Lemma 9.4 from
[17].

######  Lemma 9.11.

For $\omega$-almost every $\zeta=[p,l]$, for any $v\in E^{+}_{\zeta}$

$\|v\|\leq\|v\|_{\varepsilon_{0},-,\zeta}\leq
L(p)\|g\|_{C^{1}}e^{\varepsilon_{0}}(1-e^{-\varepsilon_{0}})^{\frac{1}{2}}\|v\|.$

In particular, by defining
$\hat{L}(\zeta)=L(p)\|g\|_{C^{1}}e^{\varepsilon_{0}}(1-e^{-\varepsilon_{0}})^{\frac{1}{2}}$,
we have

$\hat{L}(\Phi_{t}(\zeta))\leq e^{2\varepsilon_{0}(|t|+1)}\hat{L}(\zeta)$

and

$\frac{1}{\hat{L}(\zeta)}\|D\Phi_{t}(\zeta)|_{E^{+}_{\zeta}}\|\leq\|D\Phi_{t}(\zeta)|_{E^{+}_{\zeta}}\|_{\varepsilon_{0},-}\leq
e^{2\varepsilon_{0}(|t|+1)}\hat{L}(\zeta)\|D\Phi_{t}(\zeta)|_{E^{+}_{\zeta}}\|.$
(43)

####  9.4.2 The reparametrized flow and the Martingale convergence argument

From the partition $\xi^{uu}$ we may consider the partition $\tilde{\xi}^{uu}$
obtained by the sets of the form $[\xi,\\{l\\}]$ in $M$, where
$\xi\in\xi^{uu}$ and $l\in[0,1)$. This forms an $\omega$-measurable partition.
For each $\xi\in\xi^{uu}$ fix $\xi^{uu}_{p}\in\xi$ an Oseledets regular point
for $\mu$ and for $\tilde{\xi}=[\xi,l]\in\mathcal{P}$ let
$\zeta_{\tilde{\xi}}$ be the point $[p_{\xi},l]$. Given two points
$\zeta=[p,l]$ and $\eta=[q,l]$ such that $q\in W^{uu}(p)$, we write
$H^{u}_{\zeta,\eta}$ as the map induced by $H^{u}_{p,q}$ in the first
coordinate and fixing the $l$ coordinate.

Consider the $\omega$-measurable bundle $V$ over $M$ such that for
$\omega$-almost every point $\zeta$, the fiber $V_{\zeta}$ is given by
$E^{+}_{\tilde{\xi}^{uu}_{\zeta}}$. This bundle can be obtained from the
bundle $E^{+}$ over $M$ in the following way: for each $\zeta\in M$ we
identify $E^{+}_{\zeta}$ with $E^{+}_{\tilde{\xi}^{uu}_{\zeta}}$ using the
holonomy $DH^{u}_{\zeta,\tilde{\xi}^{uu}_{\zeta}}(\zeta)$, recall that $E^{+}$
is $DH^{u}$-invariant.On the bundle $V$ we may consider the linear cocycle
over $\Phi_{t}$ given by
$G_{t}(\zeta)v=DH^{u}_{\Phi_{t}(\zeta),\tilde{\xi}^{uu}_{\Phi_{t}(\zeta)}}(\Phi_{t}(\zeta))\circ
D\Phi_{t}(\zeta)\circ
DH^{u}_{\tilde{\xi}^{uu}_{\zeta},\zeta}(\tilde{\xi}^{uu}_{\zeta})v$, where
$v\in V_{\zeta}$.

######  Claim 9.12.

For any $t>0$, and for any two points $\zeta,\eta$ such that
$\eta\in\tilde{\xi}^{uu}(\zeta)$ it holds that $G_{-t}(\zeta)=G_{-t}(\eta)$.
In other words, $G_{-t}(.)$ is constant on elements of the partition
$\tilde{\xi}^{uu}$.

###### Proof.

Observe that

$\begin{array}[]{rcl}G_{-t}(\zeta)=\left(G_{t}(\Phi_{-t}(\zeta))\right)^{-1}&=&DH^{u}_{\Phi_{-t}(\zeta),\tilde{\xi}^{uu}_{\Phi_{-t}(\zeta)}}(\Phi_{-t}(\zeta))\circ
D\Phi_{-t}(\zeta)\circ
DH^{u}_{\tilde{\xi}^{uu}_{\zeta},\zeta}(\tilde{\xi}^{uu}_{\zeta})\\\
&=&DH^{u}_{\Phi_{-t}(\zeta),\tilde{\xi}^{uu}_{\Phi_{-t}(\zeta)}}(\Phi_{-t}(\zeta))\circ
DH^{u}_{\Phi_{-t}(\tilde{\xi}^{uu}_{\zeta}),\Phi_{-t}(\zeta)}\circ
D\Phi_{-t}(\tilde{\xi}^{uu}_{\zeta})\\\
&=&DH^{u}_{\Phi_{-t}(\tilde{\xi}^{uu}_{\zeta}),\tilde{\xi}^{uu}_{\Phi_{-t}(\zeta)}}\circ
D\Phi_{-t}(\tilde{\xi}^{uu}_{\zeta}).\end{array}$

Recall that the partition $\tilde{\xi}^{uu}$ verifies
$\Phi_{-t}(\tilde{\xi}^{uu}(\zeta))\subset\tilde{\xi}^{uu}(\Phi_{-t}(\zeta))$.
Thus, for any $\eta\in\tilde{\xi}^{uu}(\zeta)$ we have
$\tilde{\xi}^{uu}_{\Phi_{-t}(\eta)}=\tilde{\xi}^{uu}_{\Phi_{-t}(\zeta)}$ and
we conclude that $G_{-t}(\eta)=G_{-t}(\zeta)$.∎

For $\omega$-almost every $\zeta=[p,l]$ and any vector $v\in V_{\zeta}$ define

$\|v\|^{V}_{\varepsilon_{0},-,\zeta}:=\left(\|v\|_{\varepsilon_{0},-,p}^{V}\right)^{1-l}\left(\|G_{1}([p,0])v\|^{V}_{\varepsilon,-,g(p)}\right)^{l},$
(44)

where

$\|v\|^{V}_{\varepsilon_{0},-,p}:=\displaystyle\left(\sum_{j\leq
0,j\in\mathbb{Z}}\|G_{j}([p,0])v\|^{2}e^{-2\lambda^{+}j-2\varepsilon_{0}|j|}\right)^{\frac{1}{2}}.$

Define

$\kappa_{\zeta}(t):=\log\|G_{t}(\zeta)\|^{V}_{\varepsilon_{0},-}.$ (45)

Observe that $\|v\|_{\varepsilon_{0},-,\zeta}^{V}$ is a one-sided Lyapunov
norm for the linear cocycle $G_{t}$. From the construction of $G_{t}$ it is
easy to see that it the Lyapunov exponent of $G_{t}$ is $\lambda^{+}$. In
particular, for every $t\in\mathbb{R}$ we have the estimate

$t\lambda^{+}-|t|\varepsilon_{0}\leq\log\|G_{t}(\zeta)\|^{V}_{\varepsilon_{0},-}.$

We conclude that $\kappa_{\zeta}(.)$ is an increasing homeomorphism of
$\mathbb{R}$. Moreover, the function $\kappa_{\zeta}$ verifies the cocycle
condition, that is,
$\kappa_{\zeta}(t_{1}+t_{2})=\kappa_{\Phi_{t_{1}}(\zeta)}(t_{2})+\kappa_{\zeta}(t_{1})$.

######  Claim 9.13.

There exists a uniform constant $\hat{C}>1$ such that for $\omega$-a.e.
$\zeta$ and any $s\in\mathbb{R}$,

$\frac{1}{\hat{C}}e^{s}\leq\|D\Phi_{\kappa_{\zeta}^{-1}(s)}(\zeta)|_{E^{+}_{\zeta}}\|_{\varepsilon_{0},-}\leq\hat{C}e^{s}.$

###### Proof.

Let $t=\kappa^{-1}_{\zeta}(s)$ and observe that by definition
$\log\|G_{t}(\zeta)\|^{V}_{\varepsilon_{0},-}=s$. Fix a non-zero vector $v\in
V_{\zeta}$, hence

$\displaystyle\frac{\|G_{t}(\zeta)v\|^{V}_{\varepsilon_{0},-,\Phi_{t}(\zeta)}}{\|v\|^{V}_{\varepsilon_{0},-,\zeta}}=e^{s}.$

Set $v^{+}=DH^{u}_{\tilde{\xi}^{uu}_{\zeta},\zeta}v$ and observe that

$\|D\Phi_{t}(\zeta)|_{E^{+}_{\zeta}}\|_{\varepsilon_{0},-}=\displaystyle\frac{\|D\Phi_{t}(\zeta)v^{+}\|_{\varepsilon_{0},-,\Phi_{t}(\zeta)}}{\|v^{+}\|_{\varepsilon_{0},-,\zeta}}.$

Recall that for any $t^{\prime}\in\mathbb{R}$,
$G_{t^{\prime}}(\zeta)v=DH^{u}_{\Phi_{t^{\prime}}(\zeta),\tilde{\xi}^{uu}_{\Phi_{t^{\prime}}(\zeta)}}(\Phi_{t^{\prime}}(\zeta))\circ
D\Phi_{t^{\prime}}(\zeta)\circ
DH^{u}_{\tilde{\xi}^{uu}_{\zeta},\zeta}(\tilde{\xi}^{uu}_{\zeta})v$. In
particular,
$G_{t}(\zeta)v=DH^{u}_{\Phi_{t}(\zeta),\tilde{\xi}^{uu}_{\Phi_{t}(\zeta)}}(\Phi_{t}(\zeta))\circ
D\Phi_{t}(\zeta)v^{+}$.

Since the distance between any point $q$ and $\xi^{uu}_{q}$ is uniformly
bounded from above, there exists a uniform constant $K>1$ such that for any
non positive integer $j\leq 0$,

$\frac{1}{K}\leq\frac{\|Dg^{j}(p)v^{+}\|}{\|G_{j}([p,0])v\|}\leq K.$

The result then follows easily from the remarks above and the definitions of
the norms $\|.\|^{V}_{\varepsilon,-,\zeta}$ and
$\|.\|_{\varepsilon_{0},-,\zeta}$. ∎

Recall that $Y$ is the set of $\omega$-full measure of Oselede ts regular
points. We consider the reparametrized flow $\Psi_{s}:Y\to Y$, defined by

$\Psi_{s}(\zeta)=\Phi_{\kappa_{\zeta}^{-1}(s)}(\zeta).$

For $\zeta=[p,l]\in Y$, let
$h(\zeta)=h(\xi^{uu}_{p}):=\log\|G_{1}([p,0])v\|^{V}_{\varepsilon_{0},-,\Phi_{1}(\zeta)}$,
where $\|v\|^{V}_{\varepsilon_{0},-,\zeta}=1$. By (44), for any $t\in[-l,1-l)$
we have that

$\kappa_{\zeta}(t)=th(\zeta).$ (46)

For $s\in\mathbb{R}$ such that $\frac{s}{h(\zeta)}+l\in[0,1)$, we have that
$\Psi_{s}(\zeta)=[p,l+\frac{s}{h(\zeta)}]$. That is,
$\frac{1}{h(\xi^{uu}_{\zeta})}$ gives the local change of speed of the flow
$\Phi_{t}$ to obtain the flow $\Psi_{s}$. In particular,
$\kappa_{\zeta}(t)=\int_{0}^{t}h(\Phi_{\tau}(\zeta))d\tau$. Observe that
$h(\zeta)>\lambda^{+}-\varepsilon_{0}$ for $\omega$-almost every $\zeta$. We
also have that $\int h(\zeta)d\omega(\zeta)<+\infty$ (see Claim 9.5 in [17])
and hence the flow $\Psi_{s}$ preserves the probability measure
$\hat{\omega}$, which in coordinates is given by

$\displaystyle d\hat{\omega}(\zeta):=\frac{h(\zeta)}{\int
h(\eta)d\omega(\eta)}d\omega(\zeta).$ (47)

Since the measure $\omega$ is ergodic for $\Phi_{t}$, we obtain that
$\hat{\omega}$ is ergodic for $\Psi_{s}$.

Observe that the partition $\tilde{\xi}^{uu}$ is both $\omega$ and
$\hat{\omega}$ measurable. Let $\tilde{\mathcal{B}}$ be the $\sigma$-algebra
generated by the partition $\tilde{\xi}^{uu}$. Let
$\omega^{\tilde{\mathcal{B}}}_{\zeta}$ and
$\hat{\omega}^{\tilde{\mathcal{B}}}_{\zeta}$ be the conditional measures of
$\omega$ and $\hat{\omega}$ with respect to the $\sigma$-algebra
$\tilde{\mathcal{B}}$. This is the same as considering the disintegrated
measures of $\omega$ and $\hat{\omega}$ with respect to the measurable
partition $\tilde{\xi}^{uu}$. From (47) we obtain that

$d\hat{\omega}^{\tilde{\mathcal{B}}}_{\zeta}(\eta)=\frac{h(\eta)}{\int
h(\rho)d\omega^{\tilde{\mathcal{B}}}_{\zeta}(\rho)}d\omega^{\tilde{\mathcal{B}}}_{\zeta}(\eta).$

By construction, the function $h$ is $\tilde{\mathcal{B}}$-measurable, since
it is constant on elements of the partition $\tilde{\xi}^{uu}$. Since
$\omega^{\tilde{\mathcal{B}}}_{\zeta}$ and
$\hat{\omega}^{\tilde{\mathcal{B}}}_{\zeta}$ are probability measures, we take

$\hat{\omega}^{\tilde{\mathcal{B}}}_{\zeta}=\omega^{\tilde{\mathcal{B}}}_{\zeta}.$
(48)

The function $(\zeta,-t)\mapsto\kappa_{\zeta}(-t)$, where $t\geq 0$, is
$\tilde{\mathcal{B}}$-measurable. This follows from Claim 9.12. In particular,
the semiflow $\Psi_{-s}$ is $\tilde{\mathcal{B}}$ measurable. We also have
that $\Psi_{s}(\tilde{\mathcal{B}})\subset\mathcal{\tilde{\mathcal{B}}}$ for
$s\geq 0$, where
$\Psi_{s}(\tilde{\mathcal{B}}):=\\{\Psi_{s}(C):C\in\tilde{\mathcal{B}}\\}$,
this follows from the fact that $\tilde{\xi}^{uu}$ is decrasing for the
measurable flow $\Psi_{s}$. Write
$\tilde{\mathcal{B}}_{s}=\Psi_{s}(\tilde{\mathcal{B}})$. We have that
$\tilde{\mathcal{B}}_{s}\subset\tilde{\mathcal{B}}_{s^{\prime}}$, for $s\geq
s^{\prime}$, and we obtain that $\\{\tilde{\mathcal{B}}_{s}\\}_{s\geq 0}$
forms a decreasing filtration. Let
$\tilde{\mathcal{B}}_{\infty}:=\bigcap_{s\geq 0}\tilde{\mathcal{B}}_{s}$.

Let $\rho:Y\to\mathbb{R}$ be a $\omega$-integrable function, in particular it
is also $\hat{\omega}$-integrable. For $\hat{\omega}$-almost every $\zeta$,
define the conditional expectation
$\mathbb{E}_{\hat{\omega}}(\rho|\tilde{\mathcal{B}}_{s})(\zeta)$ by

$\mathbb{E}_{\hat{\omega}}(\rho|\tilde{\mathcal{B}}_{s})(\zeta):=\int\rho\circ\Psi_{s}(\eta)d\hat{\omega}^{\tilde{\mathcal{B}}_{0}}_{\Psi_{-s}(\zeta)}(\eta)=\int\rho(\eta^{\prime})d((\Psi_{s})_{*}\hat{\omega}^{\tilde{\mathcal{B}}_{0}}_{\Psi_{-s}(\zeta)})(\eta^{\prime}).$

By the $\Psi_{s}$-invariance of $\hat{\omega}$ it is easy to conclude that
$(\Psi_{s})_{*}\hat{\omega}^{\tilde{\mathcal{B}}_{0}}_{\Psi_{-s}(\zeta)}=\hat{\omega}^{\tilde{\mathcal{B}}_{s}}_{\zeta}$.
Furthermore, since $\Psi_{s^{\prime}}(\tilde{\xi}^{uu})$ refines
$\Psi_{s}(\tilde{\xi}^{uu})$, whenever $s\leq s^{\prime}$, we can also
conclude that
$\mathbb{E}_{\hat{\omega}}(\mathbb{E}_{\hat{\omega}}(\rho|\tilde{\mathcal{B}}_{s})|\tilde{\mathcal{B}}_{s^{\prime}})(\eta)=\mathbb{E}_{\hat{\omega}}(\rho|\tilde{\mathcal{B}}_{s^{\prime}})(\eta)$.
Thus, $\mathbb{E}_{\hat{\omega}}(\rho|\tilde{\mathcal{B}}_{s})(.)$ defines a
reverse martingale for the decreasing filtration
$\\{\tilde{\mathcal{B}}_{s}\\}_{s\geq 0}$ on $(Y,\hat{\omega})$. By the
Reverse Martingale Convergence Theorem (see [28] Theorem $5.8$) we obtain that
for $\hat{\omega}$-almost every $\zeta$ we have the convergence
$\lim_{s\to+\infty}\mathbb{E}_{\hat{\omega}}(\rho|\tilde{\mathcal{B}}_{s})(\zeta)=\mathbb{E}_{\hat{\omega}}(\rho|\tilde{\mathcal{B}}_{\infty})(\zeta)$.

###  9.5 Stopping time and bi-Lipschitz estimate

Let $\zeta\in Y$. For $\delta>0$ and $t\in\mathbb{R}$, define

$\tau_{\zeta,\delta}(t):=\sup\left\\{t^{\prime}\in\mathbb{R}:\|D\Phi_{t}(\zeta)|_{E^{-}_{\zeta}}\|^{-}_{\varepsilon_{0},\pm,\Phi_{t}(\zeta)}.\|D\Phi_{t^{\prime}}(\Phi_{t}(\zeta))|_{E^{+}_{\Phi_{t}(\zeta)}}\|^{+}_{\varepsilon_{0},\pm,\Phi_{t+t^{\prime}}(\zeta)}\delta\leq\delta\right\\}.$

Define $L_{\zeta,\delta}(t):=t+\tau_{\zeta,\delta}(t)$. Observe that the
functions $\tau_{\zeta,\delta}:\mathbb{R}\to\mathbb{R}$ and
$L_{\zeta,\delta}:\mathbb{R}\to\mathbb{R}$ are increasing homeomorphisms.

######  Lemma 9.14 ([17], Lemma $9.7$).

The functions $\tau_{\zeta,\delta}$ and $L_{\zeta,\delta}$ are bi-Lipschitz
with constants uniform in $\zeta,\delta$. In particular, for $t^{\prime}\geq
0$

$\begin{array}[]{rcccl}\frac{-\lambda^{-}-3\varepsilon_{0}}{\lambda^{+}+\varepsilon_{0}}t^{\prime}&\leq&\tau_{\zeta,\delta}(t+t^{\prime})-\tau_{\zeta,\delta}(t)&\leq&\frac{-\lambda^{-}+3\varepsilon_{0}}{\lambda^{+}-\varepsilon_{0}}t^{\prime},\\\
\frac{\lambda^{+}-\lambda^{-}-2\varepsilon_{0}}{\lambda^{+}+\varepsilon_{0}}t^{\prime}&\leq&L_{\zeta,\delta}(t+t^{\prime})-L_{\zeta,\delta}(t)&\leq&\frac{\lambda^{+}-\lambda^{-}+2\varepsilon_{0}}{\lambda^{+}-\varepsilon_{0}}t^{\prime}.\end{array}$

###  9.6 Estimates for the holonomies

We will need the following estimate on the holonomies.

######  Lemma 9.15.

Suppose that $E^{uu}_{g}$ is $\theta$-Hölder for some $\theta\in(0,1)$. There
exists a constant $L>1$ such that the following holds true: given three points
$q_{2}\in\mathbb{T}^{2}$, $q_{2}^{u}\in W^{uu}_{g_{2},1}(q_{2})$, and
$q_{2}^{s}\in W^{ss}_{g_{2},loc}(q_{2})$; let $\tilde{q}_{2}$ be the unique
point of the intersection between $W^{ss}_{g_{2},loc}(q_{2}^{u})$ and
$W^{uu}_{g_{2},2}(q_{2}^{s})$, then for any $q_{1}\in\mathbb{T}^{2}$

$d\left(H^{u}_{q_{2},q_{2}^{u}}(q_{1}),H^{s}_{\tilde{q}_{2},q_{2}^{u}}\circ
H^{u}_{q_{2}^{s},\tilde{q}_{2}}\circ
H^{s}_{q_{2},q_{2}^{s}}(q_{1})\right)<Ld(q_{2},q_{2}^{s})^{\theta}.$

Observe that if the strong foliations were jointly integrable, then for any
points as above, we would have

$H^{u}_{q_{2}^{u},q_{2}}(q_{1})=H^{s}_{q_{2}^{u},\tilde{q}_{2}},$

and Lemma 9.15 would be immediate. This lemma will give a quantitative (upper)
control on the non-integrability.

###### Proof of Lemma 9.15..

By the triangular inequality, we have

$\begin{array}[]{l}d\left(H^{u}_{q_{2},q_{2}^{u}}(q_{1}),H^{s}_{\tilde{q}_{2},q_{2}^{u}}\circ
H^{u}_{q_{2}^{s},\tilde{q}_{2}}\circ H^{s}_{q_{2},q_{2}^{s}}(q_{1})\right)\\\
\leq
d\left(H^{u}_{q_{2},q_{2}^{u}}(q_{1}),H^{u}_{q_{2}^{s},\tilde{q}_{2}}\circ
H^{s}_{q_{2},q_{2}^{s}}(q_{1})\right)+d\left(H^{u}_{q_{2}^{s},\tilde{q}_{2}}\circ
H^{s}_{q_{2},q_{2}^{s}}(q_{1}),H^{s}_{\tilde{q}_{2},q_{2}^{u}}\circ
H^{u}_{q_{2}^{s},\tilde{q}_{2}}\circ H^{s}_{q_{2},q_{2}^{s}}(q_{1})\right).\\\
\end{array}$

Since the foliation $\mathcal{F}^{uu}$ is $\theta$-Hölder and the distance
between $q_{2}$ and $q_{2}^{u}$ is uniformly bounded form above, there exists
a uniform constant $K_{1}$ such that

$d\left(H^{u}_{q_{2},q_{2}^{u}}(q_{1}),H^{u}_{q_{2}^{s},\tilde{q}_{2}}\circ
H^{s}_{q_{2},q_{2}^{s}}(q_{1})\right)\
leqK_{1}d(q_{1},H^{s}_{q_{2},q_{2}^{s}}(q_{1}).$

Since the distance between $\tilde{q}_{2}$ and $q_{2}^{s}$ is uniformly
bounded form above, we have

$d\left(H^{u}_{q_{2}^{s},\tilde{q}_{2}}\circ
H^{s}_{q_{2},q_{2}^{s}}(q_{1}),H^{s}_{\tilde{q}_{2},q_{2}^{u}}\circ
H^{u}_{q_{2}^{s},\tilde{q}_{2}}\circ H^{s}_{q_{2},q_{2}^{s}}(q_{1})\right)\leq
K_{2}d(\tilde{q}_{2},q_{2}^{u}),$

where $K_{2}$ is a uniform constant.

Notice that $d(q_{1},H^{s}_{q_{2},q_{2}^{s}}(q_{1}))\leq
K_{3}d(q_{2},q_{2}^{s})$ and $d(\tilde{q}_{2},q_{2}^{u})\leq
K_{4}d(q_{2},q_{2}^{s})$. The result then follows.

∎

###  9.7 The proof of Theorem 9.1

For this section we fix
$g\in\mathrm{Sk}^{2}(\mathbb{T}^{2}\times\mathbb{T}^{2})$ and $\mu$ an ergodic
$u$-Gibbs measure for $g$ that verifies the hypothesis of Theorem 9.1. Suppose
that $\mu$ does not verify the conclusion of Theorem 9.1. By Theorem 2.26 and
Proposition 2.27 (the Invariance Principle), we have that

$h_{\mu}(g^{-1},\mathcal{F}^{ss})<h_{\nu}(g^{-1}_{2})\leq h_{\mu}(g^{-1}),$
(49)

where $\nu=(\pi_{2})_{*}\mu$ is the unique SRB of $g_{2}$.

Let $\xi^{s}$ be a $\mu$-measurable, $s$-subordinated partition, and observe
that $\xi^{s}$ is a unstable partition for $g^{-1}$ (we recall that in our
notation $\xi^{s}$ is subordinated to the two-dimensional Pesin stable
manifolds of $g$). By Ledrappier-Young’s entropy formula results (see Theorem
C’ in [42]), we have that $h_{\mu}(g^{-1})=h_{\mu}(g^{-1},\xi^{s})$. Hence,

$h_{\mu}(g^{-1},\mathcal{F}^{ss})<h_{\mu}(g^{-1},\xi^{s})$

Take $\xi^{ss}$ a measurable partition subordinated to $\mathcal{F}^{ss}$ that
refines the partition $\xi^{s}$. For $\mu$-almost every $p$ let $\mu^{ss}_{p}$
be the conditional measure of $\mu$ along $\xi^{ss}(p)$ and let $\mu^{s}_{p}$
be the conditional measure of $\mu$ along $\xi^{s}(p)$. By Ledrappier-Young’s
entropy formula we also get that for $\mu$-almost every $p$, the dimension of
the measure $\mu^{s}_{p}$ is strictly greater than the dimension of the
measure $\mu^{ss}_{p}$. Since the measure $\mu^{s}_{p}$ can be written as
$\int_{\xi^{s}(p)}\mu^{ss}_{q}d\mu^{s}_{p}(q)$ we conclude that the measure
$\mu^{s}_{p}$ is not supported on $\xi^{ss}(p)$. Moreover, for any $\delta>0$
we could have chosen these measurable partitions having its elements with
diameter smaller than $\delta$. Since we are assuming $\mu$ to have atomic
disintegration along the center, we conclude that for any $\delta>0$, for
$\mu$-almost every $p$, there is a point $q\in W^{s}_{\delta}(p)$ such that
$q\notin W^{ss}(p)$ and $q$ is an atom of $\mu^{c}_{q}$.

####  9.7.1 Fixing several parameters and sets

We now fix the choices of several parameters to obtain a set of large measure
of “good” points for which we can apply the strategy.

  1. (A)

Fix $\beta\in(0,1)$ small such that
$\frac{1+\beta}{1-\beta}<\frac{\lambda^{+}-\lambda^{-}-2\varepsilon_{0}}{-\lambda^{-}+\varepsilon_{0}}$.

  2. (B)

Fix
$\kappa_{1}=\frac{\lambda^{+}-\lambda^{-}-2\varepsilon_{0}}{\lambda^{+}+\varepsilon_{0}}$,
$\kappa_{2}=\frac{\lambda^{+}-\lambda^{-}+2\varepsilon_{0}}{\lambda_{+}-\varepsilon_{0}}$
and $\alpha_{0}=\frac{\kappa_{1}}{5(\kappa_{1}+\kappa_{2})}$.

  3. (C)

Recall that in Section 9.4 we defined the equivalent measures $\omega$ and
$\hat{\omega}$ which are invariant for the suspension flow $\Phi_{t}$ and the
reparametrized suspension flow $\Psi_{s}$, respectively. We were also using
the notation $\zeta=[p,l]$ for points in $M$, where $p\in\mathbb{T}^{4}$ and
$l\in[0,1)$.

Fix $N_{0}>1$ large such that

$\omega\left\\{\zeta:N_{0}^{-1}\leq\frac{d\hat{\omega}}{d\omega}(\zeta)\leq
N_{0}.\right\\}>1-\frac{\alpha_{0}}{2}.$

On $M$ we may consider the measurable partition induced by the center
foliation on $\mathbb{T}^{4}$. For the measure $\omega$, we will write
$\omega^{c}_{\zeta}$ the conditional measure on the leaf containing $\zeta$.

  1. (D)

By Luzin’s Theorem we fix a compact set $K_{0}\subset Y\subset M$ of $\omega$
and $\hat{\omega}$ measure arbitrarily close to one ( where $Y$ is the set of
full $\omega$-measure defined at the beginning of Section 9.4) such that:

    1. (i)

the vector fields $\zeta\mapsto v^{*}_{\zeta}$;

    2. (ii)

the parametrizations defined in (36) and extended to $Y$ in Section 9.4;

    3. (iii)

the Lyapunov norms defined in (42);

    4. (iv)

the map $\zeta\mapsto\omega^{c}_{\zeta}$;

    5. (v)

the function $\hat{L}(.)$ from Lemma 9.6

vary continuously in $K_{0}$. In particular, $\hat{L}(.)$ is bounded by a
constant $\hat{L}_{0}$ in $K_{0}$.

Recall that by assumption $\omega_{\zeta}^{c}$ is atomic for $\omega$-almost
every $\zeta$. By the continuity of $\zeta\mapsto\omega^{c}_{\zeta}$ in
$K_{0}$, there exists a constant $\varepsilon_{1}>0$ such that for any
$\zeta\in K_{0}$

$\min\\{d(\eta_{1},\eta_{2}):\textrm{ $\eta_{1}$ and $\eta_{2}$ are different
atoms of $\omega^{c}_{\zeta}$}\\}>\varepsilon_{1}.$ (50)

  2. (E)

Let $C_{0}>1$ be the maximal ratio between the Lyapunov norms defined in (42)
and the Riemannian norm for the points in $K_{0}$, that is,

$C_{0}=\displaystyle\sup_{\zeta\in K_{0}}\sup_{v\in
T_{\zeta}\mathbb{T}^{2}_{\zeta}-\\{0\\}}\left\\{\left(\frac{\|v\|_{\varepsilon_{0},\pm,\zeta}}{\|v\|_{\varepsilon,-,\zeta}}\right)^{\pm
1},\left(\frac{\|v\|_{\varepsilon_{0},\pm,\zeta}}{\|v\|}\right)^{\pm
1},\left(\frac{\|v\|_{\varepsilon_{0},-,\zeta}}{\|v\|}\right)^{\pm
1}\right\\}.$

  3. (F)

From Remark 9.7, fix $\gamma_{1},\gamma_{2}>0$ small such that
$\mu(\mathcal{A}_{\gamma_{1},\gamma_{2},0.9})>1-\alpha_{0}$. Let
$\Lambda^{\prime}\subset\mathbb{T}^{4}$, and the constants
$C_{1},C_{2},C_{3}>1$ and $r_{0},r_{1}>0$ small given by Lemma 9.9. We may
also suppose that $\mu(\Lambda^{\prime})>1-2\alpha$. Write
$\mathpzc{A}:=\mathcal{A}_{\gamma_{1},\gamma_{2},0.9}\times[0,1)$ and for
$\zeta=[p,l]$ write
$\mathscr{A}_{\gamma_{2}}(\zeta)=\mathscr{A}_{\gamma_{2}}(p)$, where
$\mathscr{A}_{\gamma_{2}}(p)$ is defined as in Section 9.3.

  4. (G)

Let $L^{*}$ be a constant such that for any $q\in\xi^{uu}(p)$ and for any
$x,y\in\mathbb{T}^{2}$ we have

$\frac{1}{L^{*}}d(x,y)\leq d(H^{u}_{p,q}(x),H^{u}_{p,q}(y))\leq L^{*}d(x,y),$

and let $L$ be the constant obtained in Lemma 9.15. Take $C^{*}=L^{*}+L$.

  5. (H)

Take
$\hat{T}:=\displaystyle\frac{\log(\hat{C}^{2}C_{0}^{2}\hat{L}_{0}^{2}C_{3}^{3})}{\lambda^{+}-\varepsilon_{0}}$,
where $\hat{C}$ is the constant from Claim 9.13.

  6. (I)

Fix $K=K_{0}\cap[\Lambda^{\prime}\times[0,1)]$. By taking the previous sets
with sufficiently large measure, we may suppose that
$\omega(K)>1-\frac{\alpha_{0}}{10}$ and
$\hat{\omega}(K)>1-\frac{\alpha_{0}}{20N_{0}}$.

Observe that if $\rho:Y\to[0,1]$ is an integrable function such that $\int\rho
d\omega>1-ab$, for constants $a,b$ then $\omega(\\{p\in
Y:\rho(p)>1-a\\})>1-b$. Indeed, write $B=\\{p\in Y:\rho(p)>1-a\\}$, then

$1-ab<\int\rho d\omega=\int_{B}\rho d\omega+\int_{Y-B}\rho
d\omega\leq\omega(B)+(1-a)(1-\omega(B)),$

and this implies that $\omega(B)>1-b$.

Recall that in section 9.4.2 we defined
$\tilde{\mathcal{B}}_{s}=\Psi_{s}(\tilde{\mathcal{B}})$. Let
$\mathds{1}_{K}(.)$ be the indicator function of $K$.Consider
$\mathbb{E}_{\omega}(\mathds{1}_{K}|\tilde{\mathcal{B}}_{0})(\zeta)=\int\mathds{1}(\eta)d\omega^{\tilde{\mathcal{B}}_{0}}_{\zeta}(\eta)$.
Take $a=0.1$ and $b=\alpha_{0}$, we have that

$\displaystyle
1-\frac{\alpha}{10}<\omega(K)=\int\left(\int\mathds{1}_{K}(\eta)d\omega^{\tilde{\mathcal{B}}_{0}}_{\zeta}(\eta)\right)d\omega(\zeta).$

By the argument above, we conclude that $\omega(\\{\zeta\in
M:\mathbb{E}_{\omega}(\mathds{1}_{K}|\tilde{\mathcal{B}}_{0})(\zeta)>0.9\\})>1-\alpha_{0}$.

Similarly, using (I) and by taking $a=0.1$ and $b=\frac{\alpha_{0}}{2N_{0}}$,
we conclude that $\hat{\omega}(\\{\zeta\in
M:\mathbb{E}_{\hat{\omega}}(\mathds{1}_{K}|\tilde{\mathcal{B}}_{\infty})(\zeta)>0.9\\})>1-\frac{\alpha_{0}}{2N_{0}}$.
From (C), we can conclude that $\omega(\\{\zeta\in
M:\mathbb{E}_{\hat{\omega}}(\mathds{1}_{K}|\tilde{\mathcal{B}}_{\infty})(\zeta)>0.9\\})>1-\alpha_{0}$.

  1. (I)

Take $\mathfrak{B}_{0}:=\\{\zeta\in
M:\omega^{\tilde{\mathcal{B}}_{0}}_{\zeta}(K)>0.9\\}$. From the argument
above, $\omega(\mathfrak{B}_{0})>1-\alpha_{0}$.

  2. (J)

For each $N>0$ let $\mathrm{B}_{N}:=\\{\zeta\in
M:\mathbb{E}_{\hat{\omega}}(\mathds{1}_{K}|\tilde{\mathcal{B}}_{s})(\zeta)>0.9,\forall
s\geq N\\}=\\{\zeta\in
M:\hat{\omega}^{\tilde{\mathcal{B}}}_{\Psi_{-s}(\zeta)}(\Psi_{-s}(K))>0.9,\forall
s\geq N\\}$. By the Martingale convergence argument from section 9.4.2, we
have that for $\hat{\omega}$-almost every $\zeta$,

$\lim_{s\to+\infty}\mathbb{E}_{\hat{\omega}}(\mathds{1}_{K}|\tilde{\mathcal{B}}_{s})(\zeta)=\mathbb{E}_{\hat{\omega}}(\mathds{1}_{K}|\tilde{\mathcal{B}}_{\infty})(\zeta).$

Hence, fix $N$ sufficiently large so that
$\min\\{\hat{\omega}(\mathrm{B}_{N}),\omega(\mathrm{B}_{N})\\}>1-\alpha_{0}$.

For each $T>0$, let $\mathcal{R}(T)$ be the set of points $\zeta\in K$ such
that for $B=K,\mathpzc{A},\mathrm{B}_{N},\mathfrak{B}_{0}$ it holds that

$\frac{1}{T}\mathrm{Leb}(\\{t\in[0,T]:\Phi_{t}(\zeta)\in B\\})>1-\alpha._{0}$

  1. (K)

By the pointwise ergodic theorem, fix $T_{0}>0$ large enough such that
$\omega(\mathcal{R}(T_{0}))>0$.

###  9.8 Back to the proof

Recall that we supposed that $\mu$ does not verify the conclusion of Theorem
9.1. Recall that $\varepsilon_{1}>0$ is a small constant fixed in (50). Since
$\mathcal{R}(T_{0})>0$, we may fix two points in $K$, $\zeta=[p,l]$ and
$\eta=[q,l]$, such that

  * •

$\zeta,\eta\in\mathcal{R}(T_{0})$;

  * •

$q\in W^{s}_{loc}(p)$;

  * •

$q\notin W^{ss}(p)$.

Let $\delta=\|h^{-}_{p}(H^{s}_{q,p}(q))\|$, where $h^{-}_{p}$ is the local
chart obtained in Proposition 9.4 . Furthermore, we may assume that

$\delta<\frac{\varepsilon_{1}}{2C^{*}\hat{C}^{2}C_{0}^{6}C_{1}^{3}C_{2}}.$

######  Lemma 9.16.

There exists a sequence $(t_{j})_{j\in\mathbb{N}}$ of positive numbers with
$t_{j}\to+\infty$, as $j\to+\infty$, such that

  1. 1.

$\Phi_{t_{j}}(\zeta)\in K\cap\mathfrak{B}_{0}\cap\mathpzc{A}$;

  2. 2.

$\Phi_{t_{j}}(\eta)\in K\cap\mathfrak{B}_{0}$;

  3. 3.

$\Phi_{L_{\zeta,\delta}(t_{j})}(\zeta)\in K\cap\mathrm{B}_{N}$, where
$\mathrm{B}_{N}$ is defined in (J);

  4. 4.

$\Phi_{L_{\zeta,\delta}(t_{j})}(\eta)\in K\cap\mathrm{B}_{N}$.

The proof of Lemma 9.16 is the same as Claim $12.2$ in [17].

Let $t_{j}\to+\infty$ be a sequence verifying Lemma 9.16. Let
$\zeta_{j}=\Phi_{t_{j}}(\zeta)$, $\eta_{j}=\Phi_{t_{j}}(\eta)$,
$\tilde{\zeta}_{j}=\Phi_{L_{\zeta,\delta}(t_{j})}(\zeta)$ and
$\tilde{\eta}_{j}=\Phi_{L_{\zeta,\delta}(t_{j})}(\eta)$. Let
$s^{\prime}_{j}=\kappa_{\zeta_{j}}(\tau_{\zeta,\delta}(t_{j}))$ and
$s_{j}^{\prime\prime}=\kappa_{\eta_{j}}(\tau_{\zeta,\delta}(t_{j}))$. Notice
that for $t_{j}$ sufficiently large, since
$\tau_{\zeta,\delta}(t_{j})\to\infty$, and by the definition of the $\kappa$
function, we have that

$\min\\{s^{\prime}_{j},s_{j}^{\prime\prime}\\}\geq(\lambda^{+}-\varepsilon_{0}).\tau_{\zeta,\delta}(t_{j})\geq
N.$

Since $\tilde{\zeta}_{j}$, $\tilde{\eta}_{j}\in\mathrm{B}_{N}$, we have that

$\mathbb{E}_{\hat{\omega}}(\mathds{1}_{K}|\tilde{\mathcal{B}}_{s^{\prime}_{j}})(\tilde{\zeta}_{j})>0.9\textrm{
and
}\mathbb{E}_{\hat{\omega}}(\mathds{1}_{K}|\tilde{\mathcal{B}}_{s_{j}^{\prime\prime}})(\tilde{\eta}_{j})>0.9.$
(51)

For a point $\hat{\zeta}=[\hat{p},l]$, by construction, there is a natural
identification of $\omega^{\tilde{\mathcal{B}}}_{\hat{\zeta}}$ with
$\mu^{uu}_{\hat{p}}$. We also recall that in the skew product setting, since
$\mu$ is $u$-Gibbs,
$(\pi_{2})_{*}\mu^{uu}_{\hat{p}}=\nu^{uu}_{\pi_{2}(\hat{p})}$. Write
$\zeta_{j}=[p_{j},l_{j}]$ and $\eta_{j}=[q_{j},l_{j}]$. We have the following:

  * •

Since $\zeta_{j}$ and $\eta_{j}$ belong to $\mathfrak{B}_{0}$, we have that
$\min\\{\omega^{\tilde{\mathcal{B}}}_{\zeta_{j}}(K),\omega^{\tilde{\mathcal{B}}}_{\eta_{j}}(K)\\}>0.9$.

  * •

Since $\zeta_{j}\in\mathpzc{A}$, we obtain that
$\omega^{\tilde{\mathcal{B}}}_{\zeta_{j}}(\mathcal{A}_{\gamma_{2}}(\zeta_{j}))=\mu^{uu}_{p_{j}}(\mathcal{A}_{\gamma_{2}}(p_{j}))>0.9$.

  * •

Observe that
$\Psi_{-s_{j}^{\prime}}(\tilde{\zeta}_{j})=\Phi_{\kappa_{\tilde{\zeta}_{j}}^{-1}(-s_{j}^{\prime})}(\tilde{\zeta}_{j})$,
by the definition of $s^{\prime}_{j}$ and $\tilde{\zeta}_{j}$, we have that
$\Psi_{-s_{j}^{\prime}}(\tilde{\zeta}_{j})=\Phi_{-\tau_{\zeta,\delta}(t_{j})}(\tilde{\zeta}_{j})=\zeta_{j}$.
Similarly, $\Psi_{-s_{j}^{\prime\prime}}(\tilde{\eta}_{j})=\eta_{j}$.

  * •

By (51) and the previous item, we obtain that

$\hat{\omega}^{\tilde{\mathcal{B}}}_{\zeta_{j}}(\Psi_{-s_{j}^{\prime}}(K))>0.9\textrm{
and
}\hat{\omega}^{\tilde{\mathcal{B}}}_{\eta_{j}}(\Psi_{-s_{j}^{\prime\prime}}(K))>0.9.$

Using the identification in (48), we conclude that

$\omega^{\tilde{\mathcal{B}}}_{\zeta_{j}}(\Psi_{-s_{j}^{\prime}}(K))>0.9\textrm{
and
}\omega^{\tilde{\mathcal{B}}}_{\eta_{j}}(\Psi_{-s_{j}^{\prime\prime}}(K))>0.9.$

Observe that $\zeta_{j}$ and $\eta_{j}$ both have the same time coordinate.
Recall that the invariant foliations for $\Phi_{t}$ are induced by the
foliations of $g$. In particular, for any two points with the same
$l$-coordinate we can look at the holonomy map induced by the center stable
foliation between the pieces of strong unstable manifolds. We remark that the
center stable holonomy induces a $C^{1}$ map between strong unstable
manifolds. In particular, for $j\in\mathbb{N}$ sufficiently large, we can
choose points $\overline{\zeta}_{j}\in\tilde{\xi}^{uu}(\zeta_{j})$ and
$\neq{\eta}_{j}\in\tilde{\xi}^{uu}(\eta_{j})$ such that

  * •

$\overline{\eta}_{j}\in W^{cs}(\overline{\zeta}_{j})$;

  * •

$\overline{\eta}_{j}\in\Psi_{-s_{j}^{\prime\prime}}(K)\cap K$;

  * •

$\overline{\zeta}_{j}\in\Psi_{-s_{j}^{\prime}}(K)\cap
K\cap\mathcal{A}_{\gamma_{2}}(\zeta_{j})$.

Write $\overline{\zeta}_{j}=[\overline{p}_{j},l_{j}]$ and
$\overline{\eta}_{j}=[\overline{q}_{j},l_{j}]$.

Let $\overline{w}_{j}=W^{+}_{r_{1}}(\overline{p}_{j})\cap
H^{s}_{\overline{q}_{j},\overline{p}_{j}}(W^{-}_{r_{1}}(\overline{q}_{j}))$
and let $\overline{\omega}_{j}=[\overline{w}_{j},l_{j}]$. Let
$\overline{z}_{j}=H^{u}_{p_{j},\overline{p}_{j}}(W^{-}_{r_{1}}(p_{j}))\cap
H^{s}_{\overline{q}_{j},\overline{p}_{j}}(W^{-}_{r_{1}}(\overline{q}_{j}))$.
Since $\overline{p}_{j}\in\mathcal{A}_{\gamma_{2}}(p_{j})$ and
$\overline{q}_{j}\in K$, by Lemma 9.9 we obtain that

$\frac{1}{C_{1}C_{2}}d(\overline{p}_{j},\overline{z}_{j})\leq\|h^{+}_{\overline{p}_{j}}(\overline{w}_{j})\|\leq
C_{1}C_{2}d(\overline{p}_{j},\overline{z}_{j}),$

observe that by taking $j$ sufficiently large and by Lemma 9.15, we may
suppose that $p_{j},\overline{p}_{j}$ and $\overline{q}_{j}$ verify the
hypothesis of Lemma 9.9. Write $q^{*}_{j}=H^{s}_{q_{j},p_{j}}(q_{j})$

######  Lemma 9.17.

For $j$ sufficiently large, we have

$\frac{1}{C^{*}}d(p_{j},q_{j}^{*})\leq
d(\overline{p}_{j},\overline{z}_{j})\leq C^{*}d(p_{j},q_{j}^{*}),$

where $C^{*}$ is the constant defined in (G).

###### Proof.

Write $\overline{q}_{j}^{*}=H^{u}_{p_{j},\overline{p}_{j}}(q_{j}^{*})$. By
Lemma 9.15 we have that

$d(H^{u}_{p_{j},\overline{p}_{j}}(q^{*}_{j}),H^{s}_{\overline{q}_{j},\overline{p}_{j}}\circ
H^{u}_{q_{j},\overline{q}_{j}}\circ H^{s}_{p_{j},q_{j}}(q^{*}_{j}))\leq
Ld((p_{j})_{2},(q_{j})_{2})^{\theta}\leq L\|Dg|_{E^{ss}}\|^{t_{j}\theta}.$
(52)

Since the angle between $H^{u}_{p_{j},\overline{p}_{j}}(W^{-}_{r_{1}}(p_{j}))$
and
$H^{s}_{\overline{q}_{j},\overline{p}_{j}}(W^{-}_{r_{1}}(\overline{q}_{j}))$
is uniformly bounded away from zero, by (52) we conclude that
$d(\overline{q}_{j}^{*},\overline{z}_{j}))\leq
C_{4}L\|Dg|_{E^{ss}}\|^{t_{j}\theta}$, for some uniform constant $C_{4}>0$.

Figure 3: Control on distances

Since $H^{u}$ is $C^{1}$ and the diameter of the elements of the partition
$\xi^{uu}$ is uniformly bounded, there exists $L^{*}>1$ such that for any
$x,y\in\mathbb{T}^{2}$,

$\frac{1}{L^{*}}d(x,y)<d(H^{u}_{p_{j},\overline{p}_{j}}(x),H^{u}_{p_{j},\overline{p}_{j}}(y))<L^{*}d(x,y).$

Hence,

$\begin{array}[]{lll}d(\overline{p}_{j},\overline{q_{j}})&\leq&d(\overline{p}_{j},\overline{q}_{j}^{*})+d(\overline{q}_{j}^{*},\overline{z}_{j})\leq
d(H^{u}_{p_{j},\overline{p}_{j}}(p_{j}),H^{u}_{p_{j},\overline{p}_{j}}(q^{*}_{j}))+C_{4}L\|Dg|_{E^{ss}}\|^{t_{j}\theta}\\\
&\leq&\displaystyle
d(p_{j},q_{j}^{*})\left(L^{*}+C_{4}L\frac{\|Dg|_{E^{ss}}\|^{t_{j}\theta}}{d(p_{j},q_{j}^{*})}\right).\end{array}$

However, $d(p_{j},q_{j}^{*})\geq m(Dg|_{E^{c}})^{t_{j}}d(p,H^{s}_{q,p}(q))$
and by the condition (32), for $j$ sufficiently large we have

$C_{4}\frac{\|Dg|_{E^{ss}}\|^{t_{j}\theta}}{d(p_{j},q_{j}^{*})}\leq\left(\frac{\|Dg|_{E^{ss}}\|^{\theta}}{m(Dg|_{E^{c}})}\right)^{t_{j}}\frac{C_{4}}{d(p,H^{s}_{q,p}(q))}<1.$

Therefore, for $j$ sufficiently large

$d(\overline{p}_{j},\overline{q}_{j})\leq
d(p_{j},q_{j}^{*})(L^{*}+L)=C^{*}d(p_{j},q_{j}^{*}).$

The proof of the lower bound is similar. ∎

Combining the estimate from Lemma 9.17 and the estimates for
$\|h^{+}_{\overline{p}_{j}}(\overline{w}_{j})\|$, we obtain

$\displaystyle\frac{1}{C^{*}C_{1}C_{2}}d(p_{j},q^{*}_{j})\leq\|h^{+}_{\overline{p}_{j}}(\overline{w}_{j})\|\leq
C^{*}C_{1}C_{2}d(p_{j},q_{j}^{*}).$

However,

$\begin{array}[]{rcl}d(p_{j},q_{j}^{*})&\leq&C_{1}\|h^{-}_{p_{j}}(q_{j}^{*})\|\leq
C_{0}C_{1}\|h^{-}_{p_{j}}(q_{j}^{*})\|_{\varepsilon_{0},\pm,p_{j}}\\\
&=&C_{0}C_{1}\|D\Phi_{t_{j}}(\zeta)|_{E^{+}_{\zeta}}\|_{\varepsilon_{0},\pm}\|h^{-}_{p}(H^{s}_{q,p}(q))\|_{\varepsilon_{0},\pm,p}\leq
C_{0}^{2}C_{1}\|D\Phi_{t_{j}}(\zeta)|_{E^{+}_{\zeta}}\|_{\varepsilon_{0},\pm}\delta.\end{array}$

Similarly, we can obtain the lower bound

$d(p_{j},q_{j}^{*})\geq\displaystyle\frac{1}{C_{0}^{2}C_{1}}\|D\Phi_{t_{j}}(\zeta)|_{E^{+}_{\zeta}}\|_{\varepsilon_{0},\pm}\delta.$

Let $t_{j}^{\prime}=\kappa_{\overline{\zeta}_{j}}^{-1}(s_{j}^{\prime})$ and
$t_{j}^{\prime\prime}=\kappa_{\overline{\eta}_{j}}^{-1}(s_{j}^{\prime\prime})$.
Observe that
$\Psi_{s_{j}^{\prime}}(\overline{\zeta}_{j})=\Phi_{t_{j}^{\prime}}(\overline{\zeta}_{j})$
and
$\Psi_{s_{j}^{\prime\prime}}(\overline{\eta}_{j})=\Phi_{t_{j}^{\prime\prime}}(\overline{\eta}_{j})$.
Write
$\overline{\omega}_{j}^{\prime}=\Phi_{t_{j}^{\prime}}(\overline{\omega}_{j})$,
$\overline{\zeta}_{j}^{\prime}:=\Phi_{t_{j}^{\prime}}(\overline{\zeta}_{j})=[p_{j}^{\prime},l_{j}^{\prime}]$,
$\overline{\eta}_{j}^{\prime}:=\Phi_{t_{j}^{\prime}}(\overline{\eta}_{j})=[q_{j}^{\prime},l_{j}^{\prime}]$
and
$\overline{\eta}_{j}^{\prime\prime}:=\Phi_{t_{j}^{\prime\prime}}(\overline{\eta}_{j})=[q_{j}^{\prime\prime},l_{j}^{\prime\prime}]$.

Recall that $\hat{C}$ is the constant given by Claim 9.13.

######  Claim 9.18.

$\displaystyle\frac{1}{\hat{C}^{2}C_{0}^{6}C_{1}^{2}C_{2}}\delta\leq\|h^{+}_{\overline{\zeta}_{j}^{\prime}}(\overline{\omega}_{j}^{\prime})\|\leq\hat{C}^{2}C_{0}^{6}C_{1}^{2}C_{2}\delta.$

###### Proof.

$\begin{array}[]{lll}\|h^{+}_{\overline{\zeta}_{j}^{\prime}}(\overline{\omega}_{j}^{\prime})\|&\leq&C_{0}\|h^{+}_{\overline{\zeta}_{j}^{\prime}}(\overline{\omega}_{j}^{\prime})\|_{\varepsilon_{0},-,\overline{\zeta}_{j}^{\prime}}=C_{0}\|D\Phi_{t_{j}^{\prime}}(\overline{\zeta}_{j})|_{E^{+}_{\overline{\zeta}_{j}}}\|_{\varepsilon_{0},-}\|h^{+}_{\overline{\zeta}_{j}}(\overline{\omega}_{j})\|_{\varepsilon_{0},-,\overline{\zeta}_{j}}\\\
&\leq&C_{0}^{2}\hat{C}e^{s^{\prime}_{j}}\|h^{+}_{\overline{\zeta}_{j}}(\overline{\omega}_{j})\|\leq
C^{*}\hat{C}C_{0}^{4}C_{1}^{2}C_{2}\|D\Phi_{t_{j}}(\zeta)|_{E^{+}_{\zeta}}\|_{\varepsilon_{0},\pm}\delta
e^{s_{j}^{\prime}}\\\
&=&C^{*}\hat{C}C_{0}^{4}C_{1}^{2}C_{2}\|D\Phi_{t_{j}}(\zeta)|_{E^{-}_{\zeta}}\|_{\varepsilon_{0},\pm}\delta\|G_{\tau_{\zeta,\delta}(t_{j})}(\zeta_{j})\|^{V}_{\varepsilon_{0},-}\\\
&\leq&C^{*}\hat{C}^{2}C_{0}^{4}C_{1}^{2}C_{2}\|D\Phi_{t_{j}}(\zeta)|_{E^{-}_{\zeta}}\|_{\varepsilon_{0},\pm}\delta\|D\Phi_{\tau_{\zeta,\delta}(t_{j})}(\zeta_{j})|_{E^{+}_{\zeta_{j}}}\|_{\varepsilon_{0},-}\\\
&\leq&C^{*}\hat{C}^{2}C_{0}^{6}C_{1}^{2}C_{2}\|D\Phi_{t_{j}}(\zeta)|_{E^{-}_{\zeta}}\|_{\varepsilon_{0},\pm}\delta\|D\Phi_{\tau_{\zeta,\delta}(t_{j})}(\zeta_{j})|_{E^{+}_{\zeta_{j}}}\|_{\varepsilon_{0},\pm}=C^{*}\hat{C}^{2}C_{0}^{6}C_{1}^{2}C_{2}\delta.\end{array}$

The proof of the lower bound is similar. ∎

Since $\overline{\zeta}_{j}^{\prime}\in K$, we have that
$\frac{1}{C_{1}}\|h^{+}_{\overline{\zeta}_{j}^{\prime}}(\overline{\omega}_{j}^{\prime})\|\leq
d(\overline{\zeta}_{j}^{\prime},\overline{\omega}_{j}^{\prime})\leq
C_{1}\|h^{+}_{\overline{\zeta}_{j}^{\prime}}(\overline{\omega}_{j}^{\prime})\|.$
Notice as well that $\overline{\omega}_{j}$ belongs to the stable manifold of
$\overline{\eta}_{j}^{\prime}$, indeed, $\overline{w}_{j}\in
H^{s}_{\overline{q}_{j},\overline{p}_{j}}(W-_{r_{1}}(\overline{q}_{j}))$. From
the definition of $t_{j}^{\prime}$, we have that $t_{j}^{\prime}\to+\infty$ as
$j$ goes to infinity. In particular,
$d(\overline{\omega}_{j}^{\prime},\overline{\eta}_{j}^{\prime})\to 0$ as $j$
increases. Hence, for $j$ large enough, we have that

$d(\overline{\omega}_{j}^{\prime},\overline{\eta}_{j}^{\prime})<\frac{1}{2C^{*}\hat{C}^{2}C_{0}^{6}C_{1}^{3}C_{2}}\delta.$
(53)

From Claim 9.18, the estimate (53) and triangular inequality, we obtain

$\displaystyle\frac{1}{2C^{*}\hat{C}^{2}C_{0}^{6}C_{1}^{3}C_{2}}\delta\leq
d(p_{j}^{\prime},H^{s}_{q_{j}^{\prime},p_{j}^{\prime}}(q_{j}^{\prime}))\leq
2C^{*}\hat{C}^{2}C_{0}^{6}C_{1}^{3}C_{2}\delta.$ (54)

######  Claim 9.19.

$|t_{j}^{\prime}-t_{j}^{\prime\prime}|<\hat{T}$, where $\hat{T}$ is the
constant given in (H).

###### Proof.

We consider two cases.

##### Case 1: $t_{j}^{\prime}\geq t_{j}^{\prime\prime}$.

As $\overline{\zeta}_{j}\in K$, from (D) and (43),

$\|G_{t_{j}^{\prime\prime}}(\overline{\zeta}_{j})\|^{V}_{\varepsilon_{0},-}\geq\frac{1}{\hat{C}}\|D\Phi_{t_{j}^{\prime\prime}}(\overline{\zeta}_{j})|_{E^{+}_{\overline{\zeta}_{j}}}\|_{\varepsilon_{0},-}\geq\frac{1}{\hat{C}\hat{L}_{0}}\|D\Phi_{t_{j}^{\prime\prime}}(\overline{\zeta}_{j})|_{E^{+}_{\zeta}}\|.$

Since $\overline{\eta}_{j}^{\prime\prime}\in K$, we have

$\|G_{t_{j}^{\prime\prime}}(\overline{\eta}_{j})\|^{V}_{\varepsilon_{0},-}\leq\hat{C}\|D\Phi_{t_{j}^{\prime\prime}}(\overline{\eta}_{j})|_{E^{+}_{\overline{\eta}_{j}}}\|_{\varepsilon_{0},-}\leq\hat{C}C_{0}^{2}\|D\Phi_{t_{j}^{\prime\prime}}(\overline{\eta}_{j})|_{E^{+}_{\overline{\eta}_{j}}}\|.$

Let $n^{\prime}=\lfloor t_{j}^{\prime}\rfloor$ be the integer part of
$t_{j}^{\prime}$, and let $n^{\prime\prime}=\lfloor
t_{j}^{\prime\prime}\rfloor$ be the integer part of $t_{j}^{\prime\prime}$.
Observe that we are assuming that $n^{\prime\prime}\leq n^{\prime}$. We also
have that $n^{\prime\prime}\geq 0$.

$\begin{array}[]{l}\displaystyle\frac{\|Dg^{n^{\prime\prime}}(\overline{q}_{j})|_{E^{+}_{\overline{q}_{j}}}\|}{\|Dg^{n^{\prime\prime}}(\overline{p}_{j})|_{E^{+}_{\overline{p}_{j}}}\|}=\displaystyle\frac{\|Dg^{n^{\prime\prime}}(\overline{q}_{j})|_{E^{+}_{\overline{q}_{j}}}\|}{\|Dg^{n^{\prime\prime}}(\overline{w}_{j})|_{T_{\overline{w}_{j}}W^{+}(\overline{p}_{j})}\|}.\frac{\|Dg^{-n^{\prime}}(\overline{p}_{j}^{\prime})|_{E^{+}_{\overline{p}_{j}^{\prime}}}\|}{\|Dg^{-n^{\prime}}(\overline{w}_{j}^{\prime})|_{T_{\overline{w}_{j}^{\prime}}W^{+}(\overline{p}_{j}^{\prime})}\|}.\frac{\|Dg^{-(n^{\prime}-n^{\prime\prime})}(\overline{w}_{j}^{\prime})|_{T_{\overline{w}_{j}^{\prime}}(W^{+}(\overline{p}_{j}^{\prime})}\|}{\|Dg^{-(n^{\prime}-n^{\prime\prime})}(\overline{p}_{j}^{\prime})|_{E^{+}_{\overline{p}_{j}^{\prime}}}\|}\\\
\hskip 78.0pt=I.II.III\end{array}$

Recall that $\overline{w}_{j}\in
H^{s}_{\overline{q}_{j},\overline{p}_{j}}(W^{-}_{r_{1}}(\overline{q}_{j}))$.
In particular, it belongs to the stable manifold of $\overline{q}_{j}$. By
item $5$ in Lemma 9.9, $I$ is bounded by $C_{3}^{-1}$ and $C_{3}$. Observe
that $-(n^{\prime}-n^{\prime\prime})$ and $-n^{\prime}$ are negative numbers.
Since $\overline{w}_{j}^{\prime}\in W^{+}_{r_{1}}(\overline{p}_{j})$, from
item $6$ of Lemma 9.9, we obtain that $II$ and $III$ are bounded by
$C_{3}^{-1}$ and $C_{3}$. Hence,

$\frac{1}{C_{3}^{3}}\leq\displaystyle\frac{\|Dg^{n^{\prime\prime}}(\overline{q}_{j})|_{E^{+}_{\overline{q}_{j}}}\|}{\|Dg^{n^{\prime\prime}}(\overline{p}_{j})|_{E^{+}_{\overline{p}_{j}}}\|}\leq
C_{3}^{3}.$ (55)

Thus,

$\begin{array}[]{lll}\hat{C}C_{0}^{2}\|D\Phi_{t_{j}^{\prime\prime}}(\overline{\eta}_{j})|_{E^{+}_{\overline{\eta}_{j}}}\|\geq\|G_{t_{j}^{\prime\prime}}(\overline{\eta}_{j})\|^{V}_{\varepsilon_{0},-}&=&\|G_{t_{j}^{\prime}}(\overline{\zeta}_{j})\|^{V}_{\varepsilon,-}\geq
e^{(\lambda^{+}-\varepsilon_{0})(t_{j}^{\prime}-t_{j}^{\prime\prime})}\|G_{t_{j}^{\prime\prime}}(\overline{\zeta}_{j})\|^{V}_{\varepsilon_{0},-}\\\
&\geq&e^{(\lambda^{+}-\varepsilon_{0})(t_{j}^{\prime}-t_{j}^{\prime\prime})}\frac{1}{\hat{C}\hat{L}_{0}}\|D\Phi_{t_{j}^{\prime\prime}}(\overline{\zeta}_{j})|E^{+}_{\zeta}\|\end{array}$

By (55),

$e^{(\lambda^{+}-\varepsilon_{0})(t_{j}^{\prime}-t_{j}^{\prime\prime})}\leq\hat{C}^{2}C_{0}^{2}\hat{L}_{0}C_{3}^{3}.$

Therefore,

$t_{j}^{\prime}-t_{j}^{\prime\prime}\leq\displaystyle\frac{\log(\hat{C}^{2}C_{0}^{2}\hat{L}_{0}C_{3}^{3})}{\lambda^{+}-\varepsilon_{0}}=\hat{T}.$

##### Case 2: $t_{j}^{\prime}\leq t_{j}^{\prime\prime}$.

Observe that, similar to the first case,

$\|G_{t_{j}^{\prime}}(\overline{\eta}_{j})\|^{V}_{\varepsilon_{0},-}\geq\frac{1}{\hat{C}\hat{L}_{0}}\|D\Phi_{t_{j}^{\prime}}(\overline{\eta}_{j})|_{E^{+}_{\overline{\eta}_{j}}}\|\textrm{
and
}\|G_{t_{j}^{\prime}}(\overline{\zeta}_{j})\|^{V}_{\varepsilon_{0},-}\leq\hat{C}C_{0}^{2}\|D\Phi_{t_{j}^{\prime}}(\overline{\zeta}_{j})|_{E^{+}_{\overline{\zeta}_{j}}}\|.$

We have

$\|D\Phi_{t_{j}^{\prime}}(\overline{\eta}_{j})|_{E^{+}_{\overline{\eta}_{j}}}\|_{\varepsilon_{0},-}\geq\frac{1}{\hat{L}_{0}}\|D\Phi_{t_{j}^{\prime}}(\overline{\eta}_{j})|_{E^{+}_{\overline{\eta}_{j}}}\|\textrm{
and
}\|D\Phi_{t_{j}^{\prime}}(\overline{\zeta}_{j})|_{E^{+}_{\overline{\zeta}_{j}}}\|_{\varepsilon_{0},-}\leq
C_{0}^{2}\|D\Phi_{t_{j}^{\prime}}(\overline{\zeta}_{j})|_{E^{+}_{\overline{\zeta}_{j}}}\|.$

We also have

$\displaystyle\frac{\|Dg^{n^{\prime}}(\overline{q}_{j})|_{E^{+}_{\overline{q}_{j}}}\|}{\|Dg^{n^{\prime}}(\overline{p}_{j})|_{E^{+}_{\overline{p}_{j}}}\|}=\frac{\|Dg^{n^{\prime}}(\overline{q}_{j})|_{E^{+}_{\overline{q}_{j}}}\|}{\|Dg^{n^{\prime}}(\overline{w}_{j})|_{T_{\overline{w}_{j}}W^{+}_{r_{1}}(\overline{p}_{j})}\|}.\frac{\|Dg^{-n^{\prime}}(\overline{p}_{j}^{\prime})|_{E^{+}_{\overline{p}_{j}^{\prime}}}\|}{\|Dg^{-n^{\prime}}(\overline{w}_{j}^{\prime})|_{T_{\overline{w}_{j}^{\prime}}W^{+}_{r_{1}}(\overline{p}_{j}^{\prime})}\|}.$

From items $5$ and $6$ of Lemma 9.9, we obtain

$\frac{1}{C_{3}^{2}}\leq\frac{\|Dg^{n^{\prime}}(\overline{q}_{j})|_{E^{+}_{\overline{q}_{j}}}\|}{\|Dg^{n^{\prime}}(\overline{p}_{j})|_{E^{+}_{\overline{p}_{j}}}\|}\leq
C_{3}^{2}.$

Hence,

$\begin{array}[]{lll}\hat{C}C_{0}^{2}\|D\Phi_{t_{j}^{\prime}}(\overline{\zeta}_{j})|_{E^{+}_{\overline{\zeta}_{j}}}\|\geq\|G_{t_{j}^{\prime}}(\overline{\zeta}_{j})\|^{V}_{\varepsilon,-}&=&\|G_{t_{j}^{\prime\prime}}(\overline{\eta}_{j})\|^{V}_{\varepsilon_{0},-}\geq
e^{(\lambda^{+}-\varepsilon_{0})(t_{j}^{\prime\prime}-t_{j}^{\prime})}\|G_{t_{j}^{\prime}}(\overline{\eta}_{j})\|^{V}_{\varepsilon_{0},-}\\\
&\geq&e^{(\lambda^{+}-\varepsilon_{0})(t_{j}^{\prime\prime}-t_{j}^{\prime})}\frac{1}{\hat{C}\hat{L}_{0}}\|D\Phi_{t_{j}^{\prime\prime}}(\overline{\zeta}_{j})|E^{+}_{\zeta}\|.\end{array}$

Therefore,

$t_{j}^{\prime\prime}-t_{j}^{\prime}\leq\frac{\log(\hat{C}^{2}C_{0}^{2}\hat{L}_{0}C_{3}^{2})}{\lambda^{+}-\varepsilon_{0}}\leq\hat{T}.$

∎

Up to taking a subsequence, we may suppose that the sequence
$\overline{\zeta}_{j}^{\prime}$ converges to a point $\hat{\zeta}_{0}$, the
sequence $\overline{\eta}_{j}^{\prime\prime}$ converges to a point
$\hat{\eta}_{1}$, the sequence $t_{j}^{\prime}-t_{j}^{\prime\prime}$ converges
to a number $\hat{t}\in[-\hat{T},\hat{T}]$ and
$\overline{\eta}_{j}^{\prime}=\Phi_{t_{j}^{\prime}-t_{j}^{\prime\prime}}(\overline{\eta}_{j}^{\prime\prime})$
converges to a point $\hat{\eta}_{0}=\Phi_{\hat{t}}(\hat{\eta}_{1})$.

Since $q_{j}^{\prime}$ belongs to the center stable leaf of $p_{j}^{\prime}$,
from (54) we obtain that

$d(\hat{\eta}_{0},\hat{\zeta}_{0})\geq\frac{1}{2C^{*}\hat{C}^{2}C_{0}^{6}C_{1}^{3}C_{2}}\delta>0.$

Observe that since $\overline{\zeta}_{j}^{\prime}\in K$ and
$\overline{\zeta}_{j}^{\prime}$ is an atom of
$\omega_{\overline{\zeta}_{j}^{\prime}}^{c}$, then $\hat{\zeta}_{0}$ is an
atom of $\omega_{\hat{\zeta}_{0}}^{c}$. Similarly, $\hat{\eta}_{1}$ is an atom
of $\omega^{c}_{\hat{\eta}_{1}}$. By the $\Phi_{t}$-invariance of $\omega$, we
obtain that
$(\Phi_{\hat{t}})_{*}\omega^{c}_{\hat{\eta}_{1}}=\omega^{c}_{\hat{\eta}_{0}}=\omega^{c}_{\hat{\zeta}_{0}}$.
Therefore, $\omega^{c}_{\hat{\zeta}_{0}}$ has an atom in $\hat{\eta}_{0}$.
However, $d(\hat{\zeta}_{0},\hat{\eta}_{0})\leq
2C^{*}\hat{C}^{2}C_{0}^{6}C_{1}^{3}C_{2}\delta<\varepsilon_{1}$ and this is a
contradiction with (50). This concludes the proof of Theorem 9.1.

##  10 Appendix A: $C^{2}$-regularity of unstable holonomies

In this appendix we prove Theorem 2.10. Let $f$ be a $C^{2+\alpha}$ absolutely
partially hyperbolic skew product of
$\mathbb{T}^{4}=\mathbb{T}^{2}\times\mathbb{T}^{2}$ and let
$\chi^{ss},\chi^{c}_{-},\chi^{c}_{+},\chi^{uu}$ be the partially hyperbolic
constants of $f$. We say that $f$ verifies the $(2,\alpha)$-center unstable
bunching condition if

$\displaystyle\left(\frac{\chi^{c}_{+}}{\chi^{c}_{-}}\right)^{2}<\chi^{uu}\textrm{
and }\frac{\chi^{c}_{+}}{(\chi^{c}_{-})^{2}}<(\chi^{uu})^{\alpha}.$ (56)

Similarly, $f$ verifies the $(2,\alpha)$-center stable bunching condition if

$\displaystyle\chi^{ss}<\left(\frac{\chi^{c}_{-}}{\chi^{c}_{+}}\right)^{2}\textrm{
and }(\chi^{ss})^{\alpha}<\frac{\chi^{c}_{-}}{(\chi^{c}_{+})^{2}}.$ (57)

If $f$ verifies condition (56) and (57) then we say that $f$ is
$(2,\alpha)$-center bunched. In this section, for any point
$p\in\mathbb{T}^{4}$ and any $n\in\mathbb{Z}$ we write $p_{n}:=f^{n}(p)$.

In this appendix, we use the $(2,\alpha)$-center unstable bunching condition
to obtain $C^{2}$-regularity of the unstable holonomy inside a center unstable
leaf. Recall that given $p$ and $q$ belonging to the same strong unstable
leaf, then there exists a well defined strong unstable holonomy map
$H^{u}_{p,q}:W^{c}(p)\to W^{c}(q)$. Since the center manifolds are
$\mathbb{T}^{2}$, we have that each unstable holonomy is a diffeomorphism of
$\mathbb{T}^{2}$. For each $R>0$, we consider the family
$\\{H^{u}_{p,q}\\}_{p\in\mathbb{T}^{4},q\in W_{R}^{uu}(p)}$. The main theorem
of the appendix is the following:

######  Theorem 10.1 (Theorem 2.10).

Let $f$ be a $C^{2+\alpha}$ absolutely partially hyperbolic skew product of
$\mathbb{T}^{4}$, and fix $R>0$. If $f$ is $(2,\alpha)$-center unstable
bunched, then $\\{H^{u}_{p,q}\\}_{p\in\mathbb{T}^{4},q\in W_{R}^{uu}(p)}$ is a
family of $C^{2}$-diffeomorphisms of $\mathbb{T}^{2}$ whose $C^{2}$-norm
varies continuously with the choices of $p$ and $q$.

It is easy to see that this theorem follows from the case that $R=1$. Observe
that the $(2,\alpha)$-center unstable bunching condition implies
$\frac{\chi^{c}_{+}}{\chi^{c}_{-}}<\chi^{uu}.$

This condition is the regular bunching condition which is sufficient to prove
that inside a center unstable manifold, the unstable holonomy is a
$C^{1}$-diffeomorphism. For each $n\in\mathbb{Z}$, for each
$p\in\mathbb{T}^{4}$ and $q\in W^{uu}_{1}(p)$ we have

$f^{n}\circ H^{u}_{p,q}=H^{u}_{p_{n},q_{n}}\circ f^{n}\textrm{ and
}Df^{n}(H^{u}_{p,q}(.))DH^{u}_{p,q}(.)=DH^{u}_{p_{n},q_{n}}(f^{n}(.))Df^{n}(.),$
(58)

where $p_{n}=f^{n}(p)$ and $q_{n}=f^{n}(q)$.

Since the center leaves are $\mathbb{T}^{2}$, all its tangent spaces have a
canonical identification with $\mathbb{R}^{2}$. In particular, we may consider
$DH^{u}_{p,q}(.)$ to be a continuous map from $\mathbb{T}^{2}$ to
$L(\mathbb{R}^{2},\mathbb{R}^{2})$, where $L(\mathbb{R}^{2},\mathbb{R}^{2})$
is the set of linear maps from $R^{2}$ to $\mathbb{R}^{2}$. Thus, the family
$\\{DH^{u}_{p,q}(.)\\}_{p\in\mathbb{T}^{4},q\in W^{uu}_{1}(p)}$ is a
continuous family that takes values on
$C^{0}(\mathbb{T}^{2},L(\mathbb{R}^{2},\mathbb{R}^{2}))$. Furthermore, there
exists an uniform constant $C\geq 1$ such that

$\|DH^{u}_{p,q}(.)-Id\|<Cd(p,q).$ (59)

Fix some constant $K>C$ and let $\mathcal{L}$ be the set defined as follows:
an element $\mathcal{L}$ is a continuous family of maps
$\\{A_{p,q}\\}_{p\in\mathbb{T}^{4},q\in W^{uu}_{1}(p)}$ that takes value on
$C^{0}(\mathbb{T}^{2},L(\mathbb{R}^{2},\mathbb{R}^{2}))$ such that
$\|A_{p,q}-Id\|<Kd(p,q)$. For simplicity, we will denote a family
$\\{A_{p,q}\\}_{p\in\mathbb{T}^{4},q\in W^{uu}_{1}(p)}$ by $\mathcal{A}$, such
that $\mathcal{A}_{p,q}(.)=A_{p,q}(.)$. We will also write the continuous
family given the derivative of the unstable holonomy just by $DH^{u}$.

Observe that $\mathcal{L}$ has a natural distance defined by

$d(\mathcal{A},\mathcal{B})=\displaystyle\sup_{p\in\mathbb{T}^{4},q\in
W^{uu}_{1}(p)}\left\\{\sup_{x\in\mathbb{T}^{2}}\|\mathcal{A}_{p,q}(x)-\mathcal{B}_{p,q}(x)\|\right\\}.$

For each $n\in\mathbb{N}$ we define $\Gamma_{n}:\mathcal{L}\to\mathcal{L}$ in
the following way: for each $p\in\mathbb{T}^{4}$ and $q\in W^{uu}_{1}(p)$,
then

$\Gamma_{n}(\mathcal{A})_{p,q}(.)=Df^{n}(H^{u}_{p_{-n},q_{-n}}(f^{-n}(.)))\mathcal{A}_{p_{-n},q_{-n}}(f^{-n}(.))Df^{-n}(.).$
(60)

By (58), for any $n\in\mathbb{N}$ the derivative of the unstable holonomy
$DH^{u}$ is $\Gamma_{n}$-invariant, that is, $\Gamma_{n}(DH^{u})=DH^{u}$. In
the next lemma we prove that it is the only element of $\mathcal{L}$ that has
this property.

######  Lemma 10.2.

For any $\mathcal{A}\in\mathcal{L}$, the limit
$\lim_{n\to+\infty}\Gamma_{n}(\mathcal{A})$ exists and it is equal to
$DH^{u}$. Moreover, $DH^{u}$ is the only element of $\mathcal{L}$ which is
$\Gamma_{n}$-invariant for every $n\in\mathbb{N}$.

###### Proof.

Let $\mathcal{A}\in\mathcal{L}$. Fix $p\in\mathbb{T}^{4}$ and $q\in
W^{uu}_{1}(p)$, and we write $H^{u}_{-n}(.)=H^{u}_{p_{-n},q_{-n}}(.)$. We will
use a similar notation for $\mathcal{A}_{p_{-n},q_{-n}}$. For any $x\in
W^{c}(p)$, we have

$\begin{array}[]{rcl}\|\Gamma_{n}(\mathcal{A})_{p,q}(x)-DH^{u}_{p,q}(x)\|&=&\displaystyle\|Df^{n}(H^{u}_{-n}(x_{-n}))\left(\mathcal{A}_{-n}(x_{-n})-DH^{u}_{-n}(x_{-n})\right)Df^{-n}(x)\|\\\
&\leq&\displaystyle\left(\frac{\chi^{c}_{+}}{\chi^{c}_{-}}\right)^{n}\|\mathcal{A}_{-n}(x_{-n})-DH^{u}_{-n}(x_{-n})\|\\\
&\leq&\displaystyle\left(\frac{\chi^{c}_{+}}{\chi^{c}_{-}}\right)^{n}\left(\|\mathcal{A}_{-n}(x_{-n})-Id\|+\|DH^{u}_{-n}(x_{-n})-Id\|\right)\\\
&\leq&\displaystyle\left(\frac{\chi^{c}_{+}}{\chi^{c}_{-}}\right)^{n}(K+C)(\chi^{uu})^{-n}d(p,q).\end{array}$

The center bunching condition implies that

$\displaystyle\frac{\chi^{c}_{+}}{\chi^{c}_{-}}(\chi^{uu})^{-1}<1.$

Hence, $\|\Gamma_{n}(\mathcal{A})_{p,q}(x)-DH^{u}_{p,q}(x)\|$ goes to zero
uniformly as $n$ goes to infinity. Since $d(p,q)\leq 1$, this estimate is
independent of the points $p$, $q$ and $x$. In other words,
$\lim_{n\to+\infty}d(\Gamma_{n}(\mathcal{A}),DH^{u})=0$. Moreover, if
$\mathcal{A}$ is $\Gamma_{n}$-invariant for every $n\in\mathbb{N}$, then
$\lim_{n\to+\infty}d(\Gamma_{n}(\mathcal{A}),DH^{u})=d(\mathcal{A},DH^{u})=0$
and thus $\mathcal{A}=DH^{u}$. ∎

For a $C^{2}$-diffeomorphism $g:\mathbb{T}^{2}\to\mathbb{T}^{2}$, we have that
$Dg(.)$ is a map that belongs to
$C^{1}(\mathbb{T}^{2},L(\mathbb{R}^{2},\mathbb{R}^{2}))$. In particular,
$D^{2}g(.)$ is a map that belongs to
$C^{0}(\mathbb{T}^{2},L(\mathbb{R}^{2},L(\mathbb{R}^{2},\mathbb{R}^{2})))$,
where $L(\mathbb{R}^{2},L(\mathbb{R}^{2},\mathbb{R}^{2}))$ is the space of
linear maps from $\mathbb{R}^{2}$ to $L(\mathbb{R}^{2},\mathbb{R}^{2})$. The
space $L(\mathbb{R}^{2},L(\mathbb{R}^{2},\mathbb{R}^{2}))$ can be identified
with the space $L^{2}(\mathbb{R}^{2},\mathbb{R}^{2})$, which is the space of
bilinear maps of $\mathbb{R}^{2}$ taking values in $\mathbb{R}^{2}$. The space
$L^{2}(\mathbb{R}^{2},\mathbb{R}^{2})$ has a norm given by

$\|B\|=\displaystyle\sup\\{\|B(u,v)\|:\|u\|=\|v\|=1\\}.$

Using this norm, we can naturally define a $C^{0}$-metric in
$C^{0}(\mathbb{T}^{2},L^{2}(\mathbb{R}^{2},\mathbb{R}^{2}))$, which gives a
$C^{1}$-metric in $C^{1}(\mathbb{T}^{2},L(\mathbb{R}^{2},\mathbb{R}^{2}))$
that we will denote it by $d^{*}_{C^{1}}(.,.)$. We remark that the space
$C^{1}(\mathbb{T}^{2},L(\mathbb{R}^{2},\mathbb{R}^{2}))$ is complete with
$d^{*}_{C^{1}}(.,.)$.

Consider the set $\mathcal{L}^{1}$ of the elements $\mathcal{A}$ of
$\mathcal{L}$ such that for each $p\in\mathbb{T}^{4}$ and $q\in W^{uu}_{1}(p)$
we have $\mathcal{A}_{p,q}(.)\in
C^{1}(\mathbb{T}^{2},L(\mathbb{R}^{2},\mathbb{R}^{2}))$ and it varies
continuously in the $C^{1}$-topology with the choices of the points $p$ and
$q$. We define the $C^{1}$-distance on $\mathcal{L}^{1}$ by

$\displaystyle
d_{C^{1}}(\mathcal{A},\mathcal{B})=\sup_{p\in\mathbb{T}^{4},q\in
W^{uu}_{1}(p)}\left\\{d^{*}_{C^{1}}(\mathcal{A}_{p,q}(.),\mathcal{B}_{p,q}(.))\right\\}.$

It is easy to see that $\mathcal{L}^{1}$ is closed for the metric $d_{C^{1}}$.
The strategy to prove Theorem 2.10 is the following: we consider the family
$Id$ in $\mathcal{L}^{1}$ which is just the identity for any choices of
$p\in\mathbb{T}^{4}$ and $q\in W^{uu}_{1}(p)$, next we consider the sequence
$\\{\Gamma_{n}(Id)\\}_{n\in\mathbb{N}}$ and we prove that this sequence is
Cauchy for the metric $d^{*}_{C^{1}}$. Then, by Lemma 10.2 we know that
$\Gamma_{n}(Id)$ converges $C^{0}$ to $DH^{u}$. However, $\Gamma_{n}(Id)$ also
converges $C^{1}$ and therefore $DH^{u}\in\mathcal{L}^{1}$, which implies that
$\\{H_{p,q}^{u}\\}_{p\in\mathbb{T}^{4},q\in W^{uu}_{1}(p)}$ is a continuous
family of $C^{2}$-diffeomorphisms.

######  Remark 10.3.

In what follows, we will use the identification of any tangent space of
$\mathbb{T}^{2}$ with $\mathbb{R}^{2}$. So that it makes sense, for any vector
$v\in\mathbb{R}^{2}$, to consider the composition $Df(x)Df(y)v$, for any two
points $x$ and $y$. Theorem 2.10 also holds for other surfaces, the main point
that will change in the proof is two include the parallel transport between
different tangent spaces of the surface, so that we can make sense of similar
compositions. This would include some extra terms in the computation presented
below, which can also be controlled to obtain the same conclusion. For
simplicity, and having our original problem in mind (perturbations of Berger-
Carrasco’s example), we will work only on $\mathbb{T}^{2}$.

###### Proof of Theorem 2.10.

As we explained in the previous paragraph, to prove Theorem 2.10 it is enough
to prove that the sequence $\\{\Gamma_{n}(Id)\\}_{n\in\mathbb{N}}$ is a Cauchy
sequence. We fix $p\in\mathbb{T}^{4}$, $q\in W^{uu}_{1}(p)$ and
$x\in\mathbb{T}^{2}$. For each $n\in\mathbb{N}$, we define
$H^{u}_{-n}:=H^{u}_{p_{-n},q_{-n}}(x_{-}n)$ and
$\Gamma_{n}:=\Gamma_{n}(Id)_{p,q}(x)$. Observe that

$\Gamma_{n}=Df^{n}(H^{u}_{-n})Df^{-n}(x)=Df^{n}(H^{u}_{-n})Df(x_{-n-1})Df^{-1}(x_{-n})Df^{-n}(x).$

By (58), for each $j=1,\cdots,n$, we have $f^{j}(H^{u}_{-n})=H^{u}_{-n+j}$.
Hence,

$\Gamma_{n+1}=Df^{n}(H^{u}_{-n})Df(H^{u}_{-n-1})Df^{-1}(x_{-n})Df^{-n}(x).$

We want to estimate $\|D\Gamma_{n+1}-D\Gamma_{n}\|$. First, let us evaluate
$D\Gamma_{n+1}$ and $D\Gamma_{n}$. In what follows, for a diffeomorphism $g$,
we will write $D^{2}g(y)[.,.]$ to represent the bilinear form of its second
derivative on the point $y$. By the chain rule and using that
$Df(x_{-n-1})Df^{-1}(x_{-n})=Id$, we obtain

$\begin{array}[]{rclr}D\Gamma_{n}[.,.]&=&D\left(Df^{n}(H^{u}_{-n})Df(x_{-n-1})Df^{-1}(x_{-n})Df^{-n}(x)\right)[.,.]&\\\
&=&D^{2}f^{n}(H^{u}_{-n})\left[DH^{u}_{-n}Df^{-n}(x).,Df^{-n}(x).\right]&\left(\mathrm{I}_{n}\right)\\\
&&+Df^{n}(H^{u}_{-n})D^{2}f(x_{-n-1})\left[Df^{-n-1}(x).,Df^{-n-1}(x).\right]&\left(\mathrm{II}_{n}\right)\\\
&&+Df^{n}(H^{u}_{-n})Df(x_{-n-1})D^{2}f^{-1}(x_{-n})\left[Df^{-n}(x).,Df^{-n}(x).\right]&\left(\mathrm{III}_{n}\right)\\\
&&+Df^{n}(H^{u}_{-n})D^{2}f^{-n}(x)[.,.]&\left(\mathrm{IV}_{n}\right)\\\
&=&\mathrm{I}_{n}+\mathrm{II}_{n}+\mathrm{III}_{n}+\mathrm{IV}_{n}.&\end{array}$

Similarly,

$\begin{array}[]{rclr}D\Gamma_{n+1}[.,.]&=&D\left(Df^{n}(H^{u}_{-n})Df(H^{u}_{-n-1})Df^{-1}(x_{-n})Df^{-n}(x)\right)[.,.]&\\\
&=&D^{2}f^{n}(H^{u}_{-n})\left[DH^{u}_{-n}Df^{-n}(x).,Df(H^{u}_{-n-1})Df^{-n-1}(x).\right]&\left(\mathrm{I}^{\prime}_{n}\right)\\\
&&+Df^{n}(H^{u}_{-n})D^{2}f(H^{u}_{-n-1})\left[DH^{u}_{-n-1}Df^{-n-1}(x).,Df^{-n-1}(x).\right]&\left(\mathrm{II}^{\prime}_{n}\right)\\\
&&+Df^{n}(H^{u}_{-n})Df(H^{u}_{-n-1})D^{2}f^{-1}(x_{-n})\left[Df^{-n}(x).,Df^{-n}(x).\right]&\left(\mathrm{III}^{\prime}_{n}\right)\\\
&&+Df^{n+1}(H^{u}_{-n-1})Df^{-1}(x_{-n})D^{2}f^{-n}(x)[.,.]&\left(\mathrm{IV}^{\prime}_{n}\right)\\\
&=&\mathrm{I}^{\prime}_{n}+\mathrm{II}^{\prime}_{n}+\mathrm{III}^{\prime}_{n}+\mathrm{IV}^{\prime}_{n}.&\end{array}$

To estimate $\|\Gamma_{n+1}-\Gamma_{n}\|$ we will separate it into four
estimates.

### The estimate for $\|\mathrm{I}^{\prime}_{n}-\mathrm{I}_{n}\|$

Let us first write the expressions for $I_{n}$ and $I_{n}^{\prime}$. In what
follows we use that $f^{j}(H^{u}_{-n})=H^{u}_{-n+j}$, for any
$j\in\mathbb{Z}$. Then,

$\displaystyle I_{n}$ $\displaystyle=\displaystyle
D^{2}f(H^{u}_{-1})\left[Df^{n-1}(H^{u}_{-n})DH^{u}_{-n}Df^{-n}(x).,Df^{n-1}(H^{u}_{-n})Df^{-n}(x).\right]$
$\displaystyle(\mathrm{\tilde{I}}_{n,1})$ $\displaystyle+\displaystyle
Df(H^{u}_{-1})D^{2}f(H^{u}_{-2})\left[Df^{n-2}(H^{u}_{-n})DH^{u}_{-n}Df^{-n}(x).,Df^{n-2}(H^{u}_{-n})Df^{-n}(x).\right]$
$\displaystyle(\mathrm{\tilde{I}}_{n,2})$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\vdots$
$\displaystyle+\displaystyle Df(H^{u}_{-1})\cdots
Df(H^{u}_{-n+1})D^{2}f(H^{u}_{-n})\left[DH^{u}_{-n}Df^{-n}(x).,Df^{-n}(x).\right].$
$\displaystyle(\mathrm{\tilde{I}}_{n,n})$

We also have

$\displaystyle I_{n}^{\prime}$ $\displaystyle=\displaystyle
D^{2}f(H^{u}_{-1})\left[Df^{n-1}(H^{u}_{-n})DH^{u}_{-n}Df^{-n}(x).,Df^{n}(H^{u}_{-n-1})Df^{-n-1}(x).\right]$
$\displaystyle(\mathrm{\tilde{I}}_{n,1}^{\prime})$
$\displaystyle+\displaystyle
Df(H^{u}_{-1})D^{2}f(H^{u}_{-2})\left[Df^{n-2}(H^{u}_{-n})DH^{u}_{-n}Df^{-n}(x).,Df^{n-1}(H^{u}_{-n-1})Df^{-n-1}(x).\right]$
$\displaystyle(\mathrm{\tilde{I}}_{n,2}^{\prime})$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\vdots$
$\displaystyle+\displaystyle Df(H^{u}_{-1})\cdots
Df(H^{u}_{-n+1})D^{2}f(H^{u}_{-n})\left[DH^{u}_{-n}Df^{-n}(x).,Df^{-n-1}(x).\right].$
$\displaystyle(\mathrm{\tilde{I}}_{n,n}^{\prime})$

Let $\tilde{C}_{n}=Df(H^{u}_{-n-1})Df^{-1}(x_{-n})-Id$. For each $j=1,\cdots
n$, we obtain

$\begin{array}[]{rcl}\|\mathrm{\tilde{I}}_{n,j}-\mathrm{\tilde{I}}_{n,j}\|&=&\displaystyle\|Df(H^{u}_{-1})\cdots
Df(H^{u}_{-j+1}).\\\ &&\displaystyle
D^{2}f(H^{u}_{-j})\left[Df^{n-j}(H^{u}_{-n})DH^{u}_{-n}Df^{-n}(x).,Df^{n-j}(H^{u}_{-n})\tilde{C}_{n}Df^{-n}(x).\right]\|\\\
&\leq&\displaystyle\|Df^{j-1}|_{E^{c}}\|\|f\|_{C^{2}}\|Df^{n-j}|_{E^{c}}\|^{2}\|DH^{u}_{-n}\|\|Df^{-n}|_{E^{c}}\|^{2}\|\tilde{C}_{n}\|\\\
&\leq&\displaystyle\|f\|_{C^{2}}\frac{(\chi^{c}_{+})^{j-1}.(\chi^{c}_{+})^{2(n-j)}}{(\chi^{c}_{-})^{2n}}\|DH^{u}_{-n}\|\|\tilde{C}_{n}\|\\\
&=&\displaystyle\|f\|_{C^{2}}\|DH^{u}_{-n}\|\left(\frac{\chi^{c}_{+}}{\chi^{c}_{-}}\right)^{2n}\|\tilde{C}_{n}\|(\chi^{c}_{+})^{-j-1}<\|f\|_{C^{2}}\|DH^{u}_{-n}\|\left(\frac{\chi^{c}_{+}}{\chi^{c}_{-}}\right)^{2n}\|\tilde{C}_{n}\|.\end{array}$

We remark that in the last inequality we used that $\chi^{c}_{+}>1$. By (59),
for every $n\in\mathbb{N}$, we have $\|DH^{u}_{-n}\|<K$, for some constant
$K\geq 1$. Also

$\begin{array}[]{rcl}\|\tilde{C}_{n}\|&=&\displaystyle\|Df(H^{u}_{-n-1})Df^{-1}(x_{-n})-Id\|\\\
&=&\displaystyle\|\left(Df(H^{u}_{-n-1})-Df(x_{-n-1})\right)Df^{-1}(x_{-n})\|\\\
&\leq&\displaystyle\frac{1}{\chi^{c}_{-}}\|f\|_{C^{2}}d(x_{-n-1},H^{u}_{-n-1})\\\
&\leq&\displaystyle\frac{1}{\chi^{c}_{-}}\|f\|_{C^{2}}(\chi^{uu})^{-n-1}d(p,q)\leq\frac{1}{\chi^{c}_{-}}\|f\|_{C^{2}}(\chi^{uu})^{-n-1}.\end{array}$

Hence,

$\displaystyle\|\mathrm{\tilde{I}}_{n,j}-\mathrm{\tilde{I}}_{n,j}\|\leq\frac{\|f\|_{C^{2}}^{2}K}{\chi^{uu}\chi^{c}_{+}\chi^{c}_{-}}\left[\left(\frac{\chi^{c}_{+}}{\chi^{c}_{-}}\right)^{2}(\chi^{uu})^{-1}\right]^{n}.$

Take the constant

$C_{1}:=\displaystyle\frac{\|f\|_{C^{2}}^{2}K}{\chi^{uu}\chi^{c}_{+}\chi^{c}_{-}}$

and observe that

$\displaystyle\|\mathrm{I}_{n}^{\prime}-\mathrm{I}_{n}\|\leq\sum_{j=1}^{n}\|\mathrm{\tilde{I}}_{n,j}-\mathrm{\tilde{I}}_{n,j}\|\leq
C_{1}\left(\frac{(\chi^{c}_{+})^{2}}{\chi^{uu}(\chi^{c}_{-})^{2}}\right)^{n}.$
(61)

This gives the estimate we need for
$\|\mathrm{I}_{n}^{\prime}-\mathrm{I}_{n}\|$.

### The estimate for $\|\mathrm{II}^{\prime}_{n}-\mathrm{II}_{n}\|$

This is the only part in the proof of Theorem 2.10 that we use that $f$ is
$C^{2+\alpha}$. Let

$\tilde{\mathrm{II}}_{n}:=D^{2}f(x_{-n-1})\left[Df^{-n-1}(x).,Df^{-n-1}(x).\right]-D^{2}f(H^{u}_{-n-1})\left[DH^{u}_{-n-1}Df^{-n-1}(x).,Df^{-n-1}(x).\right].$

Notice that

$\|\mathrm{II}^{\prime}_{n}-\mathrm{II}_{n}\|=\|Df^{n}(H^{u}_{-n})\tilde{\mathrm{II}}_{n}\|\leq(\chi^{c}_{+})^{n}\|\tilde{\mathrm{II}}_{n}\|.$

By the triangular inequality,

$\begin{array}[]{rcl}\|\tilde{\mathrm{II}}_{n}\|&\leq&\displaystyle\|D^{2}f(x_{-n-1})\left[Df^{-n-1}(x).,Df^{-n-1}(x).\right]\\\
&&\displaystyle-D^{2}f(x_{-n-1})\left[DH^{u}_{-n-1}Df^{-n-1}(x).,Df^{-n-1}(x).\right]\|\\\
&&+\displaystyle\|D^{2}f(x_{-n-1})\left[DH^{u}_{-n-1}Df^{-n-1}(x).,Df^{-n-1}(x).\right]\\\
&&\displaystyle-D^{2}f(H^{u}_{-n-1})\left[DH^{u}_{-n-1}Df^{-n-1}(x).,Df^{-n-1}(x).\right]\|\\\
&=&\|D_{n}\|+\|E_{n}\|.\end{array}$

Let us estimate each of these terms..

$\begin{array}[]{rcl}\|D_{n}\|&=&\displaystyle\left\lVert
D^{2}f(x_{-n-1})\left[\left(Id-
DH^{u}_{-n-1}\right)Df^{-n-1}(x).,Df^{-n-1}(x).\right]\right\lVert\\\
&\leq&\displaystyle\|f\|_{C^{2}}\|Id-
DH^{u}_{-n-1}\|\|Df^{-n-1}(x)|_{E^{c}}\|^{2}\\\
&\leq&\displaystyle\|f\|_{C^{2}}\left(\frac{1}{(\chi^{c}_{-})^{2}}\right)^{n+1}C(\chi^{uu})^{-n-1}d(p,q)\\\
&\leq&\displaystyle\|f\|_{C^{2}}C\left(\frac{1}{\chi^{uu}(\chi^{c}_{-})^{2}}\right)^{n+1}\leq\|f\|_{C^{2}}C\left(\frac{1}{(\chi^{uu})^{\alpha}(\chi^{c}_{-})^{2}}\right)^{n+1}.\end{array}$

Since $f$ is $C^{2+\alpha}$, There exists a constant $C_{H}\geq 1$ such that
$\|D^{2}f(z)[.,.]-D^{2}f(w)[.,.]\|\leq C_{H}d(z,w)^{\alpha}$. Recall that
$\|DH^{u}_{-j}\|<K$, for every $j\in\mathbb{N}$ and some constant $K\geq 1$.
Therefore,

$\begin{array}[]{rcl}\|E_{n}\|&\leq&C_{H}d(x_{-n-1},H^{u}_{-n-1})^{\alpha}\|DH^{u}_{-n-1}\|\|Df^{-n-1}|_{E^{c}}\|^{2}\\\
&\leq&\displaystyle
C_{H}K\frac{1}{(\chi^{c}_{-})^{2(n+1)}}(\chi^{uu})^{-\alpha(n+1)}d(p,q)\\\
&\leq&\displaystyle
C_{H}K\left(\frac{1}{(\chi^{uu})^{\alpha}(\chi^{c}_{-})^{2}}\right)^{n+1}.\end{array}$

Take the constant

$C_{2}:=\displaystyle\left(\|f\|_{C^{2}}C+C_{H}K\right)\frac{1}{(\chi^{uu})^{\alpha}(\chi^{c}_{-})^{2}}.$

We obtain

$\displaystyle\|\mathrm{II}^{\prime}_{n}-\mathrm{II}_{n}\|\leq
C_{2}\left(\frac{\chi^{c}_{+}}{(\chi^{uu})^{\alpha}(\chi^{c}_{-})^{2}}\right)^{n}.$
(62)

The $(2,\alpha)$-center bunching condition implies that the right hand side of
(62) goes exponentially fast to zero. This gives the estimate we need for
$\|\mathrm{II}^{\prime}_{n}-\mathrm{II}_{n}\|$.

### The estimate for $\|\mathrm{III}^{\prime}_{n}-\mathrm{III}_{n}\|$

Observe that

$\begin{array}[]{rcl}\displaystyle\|\mathrm{III}_{n}^{\prime}-\mathrm{III}_{n}\|&=&\displaystyle\|Df^{n}(H^{u}_{-n})\left(Df(H^{u}_{-n-1})-Df(x_{-n-1})\right)D^{2}f^{-1}(x_{-n})\left[Df^{-n}(x).,Df^{-n}(x).\right]\|\\\
&\leq&\displaystyle(\chi^{c}_{+})^{n}\|Df(H^{u}_{-n-1})-Df(x_{-n-1})\|(\chi^{c}_{-})^{-2n}.\end{array}$

We have

$\displaystyle\|Df(H^{u}_{-n-1})-Df(x_{-n-1})\|\leq\|f\|_{C^{2}}(\chi^{uu})^{-n-1}.$

By taking

$C_{3}:=\frac{\|f\|_{C}^{2}}{\chi^{uu}},$

we conclude that

$\displaystyle\|\mathrm{III}_{n}^{\prime}-\mathrm{III}_{n}\|\leq
C_{3}\displaystyle\left(\frac{\chi^{c}_{+}}{\chi^{uu}(\chi^{c}_{-})^{2}}\right)^{n}.$
(63)

This concludes the estimate we need for
$\|\mathrm{III}_{n}^{\prime}-\mathrm{III}_{n}\|$.

### The estimate for $\|\mathrm{IV}^{\prime}_{n}-\mathrm{IV}_{n}\|$

Notice that

$\begin{array}[]{rcl}\|\mathrm{IV}_{n}^{\prime}-\mathrm{IV}_{n}\|&=&\|Df^{n}(H^{u}_{-n})\left(Df(H^{u}_{-n-1})Df^{-1}(x_{-n})-Id\right)D^{2}f^{-n}(x)[.,.]\|\\\
&\leq&(\chi^{c}_{+})^{n}\|\left(Df(H^{u}_{-n-1})-Df(x_{-n-1}\right)Df^{-1}(x_{-n})\|\|D^{2}f^{-n}(x)\|\\\
&\leq&(\chi^{c}_{+})^{n}\|f\|_{C^{2}}(\chi^{uu})^{-n-1}(\chi^{c}_{-})^{-1}\|D^{2}f^{-n}(x)\|.\end{array}$

Let us estimate $\|D^{2}f^{-n}(x)\|$. First, observe that

$\displaystyle D^{2}f^{-n}(x)[.,.]$
$\displaystyle=D^{2}f^{-1}(x_{-n+1})\left[Df^{-n+1}(x).,Df^{-n+1}(x).\right]$
$\displaystyle+Df^{-1}(x_{-n+1})D^{2}f^{-1}(x_{-n+2})\left[Df^{-n+2}(x).,Df^{-n+2}(x).\right]$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\vdots$
$\displaystyle+Df^{-n+1}(x_{-1})D^{2}f(x)[.,.].$

Using that $\|D^{2}f^{-1}(.)\|\leq\|f^{-1}\|_{C^{2}}$ and by the expression
above, we obtain

$\|D^{2}f^{-n}(x)\|\leq\displaystyle\|f^{-1}\|_{C^{2}}\sum_{j=0}^{n-1}(\chi^{c}_{-})^{-j}(\chi^{c}_{-})^{-2n+2j}=\|f^{-1}\|_{C^{2}}(\chi^{c}_{-})^{-2n}\sum_{j=0}^{n-1}(\chi^{c}_{-})^{j}.$

Since $\chi^{c}_{-}<1$, the sum $\sum_{j\in\mathbb{N}}(\chi^{c}_{-})^{j}$
converges. Define the constant $C_{4}$ as

$C_{4}:=\displaystyle\frac{\|f\|_{C^{2}}\|f^{-1}\|_{C^{2}}\sum_{j\in\mathbb{N}}(\chi^{c}_{-})^{j}}{\chi^{uu}\chi^{c}_{-}}.$

We conclude that

$\displaystyle\|\mathrm{IV}_{n}^{\prime}-\mathrm{IV}_{n}\|\leq
C_{4}\left(\frac{\chi^{c}_{+}}{\chi^{uu}(\chi^{c}_{-})^{2}}\right)^{n}.$ (64)

### Conclusion of the proof of Theorem 2.10

Take

$\displaystyle\chi=\max\left\\{\frac{(\chi^{c}_{+})^{2}}{\chi^{uu}(\chi^{c}_{-})^{2}},\frac{\chi^{c}_{+}}{(\chi^{uu})^{\alpha}(\chi^{c}_{-})^{2}},\frac{\chi^{c}_{+}}{\chi^{uu}(\chi^{c}_{-})^{2}}\right\\},$

and observe that by the $(2,\alpha)$-center bunching condition $\chi<1$. Fix
the constant $\hat{C}:=C_{1}+C_{2}+C_{3}+C_{4}$. By (61),(62), (63) and (64)
we obtain that

$\|\Gamma_{n+1}-\Gamma_{n}\|\leq\hat{C}\chi^{n}.$

Therefore, $\\{\Gamma_{n}\\}_{n\in\mathbb{N}}$ is a Cauchy sequence for the
$C^{1}$-topology. Observe that all these estimates and constants are uniform
with the choices of $p\in\mathbb{T}^{4}$, $q\in W^{uu}_{1}(p)$ and $x\in
W^{c}(p)$. We conclude that $\\{\Gamma_{n}(Id)\\}_{n\in\mathbb{N}}$ is a
Cauchy sequence in $\mathcal{L}^{1}$ for the $C^{1}$-topology. Since
$\Gamma_{n}(Id)$ converges $C^{0}$ to $DH^{u}$, we conclude that $DH^{u}$ is
$C^{1}$. This implies that $\\{H^{u}_{p,q}(.)\\}_{p\in\mathbb{T}^{4},q\in
W^{uu}_{1}(p)}$ is a continuous family of $C^{2}$-diffeomorphisms whose
$C^{2}$-norm varies continuously with the choices of $p$ and $q$ as above. ∎

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