# Rigidity of pressures of Hölder potentials and the fitting of analytic
functions via them

Liangang Ma and Mark Pollicott  Liangang Ma, School of Mathematics and
Statistics, Ludong University, Yantai 264025, Shandong, P. R. China.
<EMAIL_ADDRESS>Mark Pollicott, Department of Mathematics, Warwick
University, Coventry, CV4 7AL, UK<EMAIL_ADDRESS>
###### Abstract.

The first part of this work is devoted to the study of higher differentials of
pressure functions of Hölder potentials on shift spaces of finite type. By
describing the differentials of pressure functions via the Central Limit
Theorem for the associated random processes, we discover some rigid
relationships between differentials of various orders. The rigidity imposes
obstructions on fitting candidate convex analytic functions by pressure
functions of Hölder potentials globally, which answers a question of
Kucherenko-Quas. In the second part of the work we consider fitting candidate
analytic germs by pressure functions of locally constant potentials. We prove
that all 1-level candidate germs can be realised by pressures of some locally
constant potentials, as long as number of the symbolic set is large enough.
There are also some results on fitting 2-level germs by pressures of locally
constant potentials obtained in the work.

The first author is supported by 12001056 from NSFC and ZR2019QA003 from
SPNSF. The second author is partly supported by ERC-Advanced Grant
833802-Resonances and EPSRC Grant EP/T001674/1.

##  1\. Introduction

This work deals with traditional topics in thermodynamic formalism [Bow,
Rue1], which originates from theoretical physics. We focus on shift spaces of
finite type here, which model dynamics of some smooth systems such as Axiom-A
Diffeomorphisms through Markov partitions. Given a symbolic set $\Lambda$ of
finite symbols and a continuous potential (observable) $\phi$ on the shift
space $\Lambda^{\mathbb{N}}$, a core concept in thermodynamic formalism is the
pressure $P(\phi)$. People are particularly interested in the pressure
function $P(t\phi)$ with the variable $t>0$ representing the inverse
temperature. A sharp change in the pressure function (or other terms) is
usually called a phase transition as $t$ varies, see for example [IRV, IT1,
IT2, KQW, Lop1, Lop2, Lop3, Sar].

For Hölder continuous potentials, Ruelle [Rue2] proved that the pressure
function $P(t\phi)$ is analytic for $t\in(0,\infty)$ (in fact he proved that
$P(\psi)$ depends analytically on $\psi$ for $\psi$ in the Hölder space
$C^{0,h}(X)$ with $X$ being a transitive subshift space of finite type and
$0<h\leq 1$ being the exponent [GT]). A key ingredient in his proof is the use
of Ruelle (transfer) operator [BDL, GLP] acting on functions in the Hölder
space. Moreover, the equilibrium measure of $t\phi$ for any $t>0$ and Hölder
potential $\phi$ is always unique, so there are in fact no phase transitions
in this case. Let

$P^{(n)}(t)=P^{(n)}(t\phi)=\cfrac{d^{n}P(t\phi)}{dt^{n}}$

be the $n$-th differential of the pressure function $P(t\phi)$ with respect to
$t\in(0,\infty)$ for some fixed Hölder potential $\phi$. We also write

$P^{(1)}(t)=P^{\prime}(t),P^{(2)}(t)=P^{\prime\prime}(t),P^{(3)}(t)=P^{\prime\prime\prime}(t),\cdots$

intermittently in the following. We discover that there is some rigid
relationship between the differentials of the pressure function.

######  1.1 Theorem.

For a Hölder potential $\phi$ on a full shift space of finite type, let
$P(t)=P(t\phi)$ be its pressure. Then there exists some positive number
$M_{\phi}$ depending on $\phi$, such that

(1.1) $\sqrt{2\pi^{3}}(P^{(2)}(t))^{3/2}|P^{(3)}(t)|\leq
9|P^{(3)}(t)|+2|P^{(4)}(t)|+3\sqrt{2\pi^{3}}M_{\phi}(P^{(2)}(t))^{5/2}$

for any $t>0$.

A potential $\phi$ is said to be _generic_ (or we say it defines a non-lattice
distribution, cf. [CP, Fel, PP]), if for any normalised potential $\psi$, the
spectral radius of the complex Ruelle operator $\mathcal{L}_{\psi+it\phi}$ is
less than $1$ for any $t\neq 0$. For pressure functions of generic potentials,
Theorem 1.1 can be strengthened to the following result.

######  1.2 Theorem.

For a generic Hölder potential $\phi$ on a full shift space of finite type,
let $P(t)=P(t\phi)$ be its pressure. Then there exists some positive number
$M_{\phi}$ depending on $\phi$, such that

(1.2) $|P^{(3)}(t)\big{(}1-\sqrt{2\pi}(P^{(2)}(t))^{3/2}\big{)}|\leq
3M_{\phi}P^{(2)}(t)$

for any $t>0$.

This means the second differential of the pressure function of a generic
Hölder potential imposes some global subtle restriction on its third
differential. It would be interesting to try to interpret the meaning of
$P^{\prime\prime}(t)=\frac{1}{\sqrt[3]{2\pi}}$ for the pressure function at
individual parameters. Let
$\sigma:\Lambda^{\mathbb{N}}\rightarrow\Lambda^{\mathbb{N}}$ denote the shift
map. Both the proofs of Theorem 1.1 and 1.2 require use of the Ruelle operator
and the Central Limit Theorem (CLT) for the process
$\\{f\circ\sigma^{n}\\}_{n\in\mathbb{N}}$, with the latter one depending on a
finer CLT in the generic case. Recall that there are some expressions on the
higher differentials of the pressure function by Kotani and Sunada in [KS1]
for smooth systems, and we refer the readers to [KS2] for a CLT for random
walks on crystal lattices.

It is well-known that $P(t\phi)$ is convex and Lipschitz for continuous
$\phi$, moreover, the supporting lines of its graph must intersect the
vertical axis in a closed bounded interval in $[0,\infty)$. Kucherenko and
Quas have shown that any such function can be realised by the pressure
function of some continuous potential on some shift space of finite type [KQ,
Theorem 1], whose result fits into Katok’s flexibility programme [BKR].
However, the continuous potentials constructed in their work are not Hölder,
so they ask the following question (their original problem is set in the
multidimensional case).

######  1.3 Problem (Kucherenko-Quas).

Can a convex, Lipschitz analytic function with its supporting lines
intersecting the vertical axis in a closed bounded interval in $[0,\infty)$ be
realised by the pressure function of some Hölder potential on some shift space
of finite type?

Our following results are dedicated to an answer to their problem. We first
point out that any convex, Lipschitz analytic function with its supporting
lines intersecting the vertical axis in a closed bounded interval in
$[0,\infty)$ can be approximated by sequences of pressure functions of locally
constant potentials on some shift space of finite type.

######  1.4 Corollary.

Let $F(t)$ be a convex Lipschitz function on $(\alpha,\infty)$ for some
$\alpha>0$ with Lipschitz constant $L>0$, such that its supporting lines
intersect the vertical axis in $[\underline{\gamma},\overline{\gamma}]$ with
$0\leq\underline{\gamma}\leq\overline{\gamma}<\infty$. Then there exists a
sequence of locally constant potentials $\\{\phi_{n}\\}_{n=1}^{\infty}$ on
some shift space of finite type, such that

(1.3) $\lim_{n\rightarrow\infty}P(t\phi_{n})=F(t)$

for any $t\in(\alpha,\infty)$.

###### Proof.

This is an instant corollary of Kucherenko-Quas’ result. Let

$\Lambda=\\{0,1,\cdots,\lfloor
e^{\overline{\gamma}}\rfloor\\}\times\\{\lfloor\underline{\gamma}\rfloor,\cdots,\lceil\overline{\gamma}\rceil\\}\\}\times\\{\lfloor-L\rfloor,\cdots,\lceil
L\rceil\\}$,

in which $\lfloor\ \rfloor$ and $\lceil\ \rceil$ represent the floor and
ceiling function respectively. According to [KQ, Theorem 1], there exists a
continuous potential $\phi_{F}:\Lambda^{\mathbb{Z}}\rightarrow\mathbb{R}$,
such that

$P(t\phi_{F})=F(t)$

on $(\alpha,\infty)$. Now let

$\phi_{n}(x)\doteq\phi_{n,-}(x)=\inf\\{\phi_{F}(x):x\in[x_{-n}x_{-n+1}\cdots
x_{n}]\\}$

for any $x=\cdots x_{-(n+1)}x_{-n}\cdots
x_{n}x_{n+1}\cdots\in\Lambda^{\mathbb{Z}}$ and $n\in\mathbb{N}$, in which
$[x_{-n}x_{-n+1}\cdots x_{n}]$ means the corresponding cylinder set.
$\phi_{n}$ is a locally constant potential for any fixed $n$. Now fix
$t\in(\alpha,\infty)$, by properties of the pressure function (see for example
[Rue1, 6.8]),

(1.4)
$|P(t\phi_{n})-P(t\phi_{F})|\leq|t|\parallel\phi_{n}-\phi_{F}\parallel_{\infty}.$

Since $\phi_{F}$ is continuous, this implies (1.3).

∎

One can see that in the above proof the increasing sequence of pressures
$\\{P(t\phi_{n,-})\\}_{n\in\mathbb{N}}$ satisfies

$P(t\phi_{n,-})\nearrow F(t)$

as $n\rightarrow\infty$ since $\\{\phi_{n,-}\\}_{n\in\mathbb{N}}$ is an
increasing sequence tending to $\phi_{F}$ (see [Wal1, Theorem 9.7(ii)]).
Alternatively, one can take

$\phi_{n,+}(x)=\sup\\{\phi_{F}(x):x\in[x_{-n}x_{-n+1}\cdots x_{n}]\\}$,

which results in a decreasing sequence of locally constant potentials
approximating $\phi_{F}(x)$, or

$\phi_{n,\pm}(x)=\cfrac{\phi_{n,-}(x)+\phi_{n,+}(x)}{2}$,

which also results in a sequence of locally constant potentials approximating
$\phi_{F}(x)$, while their pressure functions both approximate $F(t)$. See
Corollary 5.5 for an interpretation of the result from another point of view.

######  1.5 Remark.

The convergence in Corollary 1.4 is uniform for $t$ in a bounded domain since
$\Lambda^{\mathbb{Z}}$ is a compact metric space by considering (1.4).

######  1.6 Remark.

A locally constant potential is of course Hölder, so according to Ruelle’s
result, the pressure functions $\\{P(t\phi_{n,-})\\}_{n\in\mathbb{N}}$ are all
analytic.

The following result confirms that some convex analytic functions cannot be
fitted by the pressure of any Hölder potential on any shift space of finite
type, which gives a negative answer to Problem 1.3.

######  1.7 Theorem.

For any $\alpha>0$, there exists a strictly convex analytic function $F(t)$ on
$(\alpha,\infty)$, with its supporting lines intersecting the vertical axis in
$[\underline{\gamma},\overline{\gamma}]\subset[0,\infty)$, such that there
does not exist any Hölder potential $\phi$ on any shift space of finite type
satisfying

$P(t\phi)=F(t)$

on $(\alpha,\infty)$.

For an explicit example of convex analytic functions in Theorem 1.7, one can
simply take

$F_{2,3,1}(t)=\cfrac{2t^{2}+3t+te^{-t^{2}}+e^{-t^{2}}}{t}$

on $(\alpha,\infty)$ for any $\alpha>0$. See Proposition 4.2 for a family of
such examples. Thus one can see that there are in fact elementary functions
which cannot be fitted by pressures of Hölder potentials on shift spaces of
finite type.

In the following we consider fitting convex analytic functions locally instead
of globally, only by pressures of locally constant potentials on shift spaces
of finite type. Let

$\Lambda_{n}=\\{1,2,\cdots,n\\}$

be the symbolic set of $n$ symbols.

######  1.8 Theorem.

Let $t_{*}>0$ and $(a_{0},a_{1})\in\mathbb{R}^{2}$ satisfying

(1.5) $\frac{a_{0}}{t_{*}}>a_{1}.$

Then for any $n\in\mathbb{N}$ large enough, there exist some $0\leq
m_{t_{*},a_{0},a_{1},n}<M_{t_{*},a_{0},a_{1},n}<\infty$ depending on
$t_{*},a_{0},a_{1},n$, such that for any
$a_{2}\in[m_{t_{*},a_{0},a_{1},n},M_{t_{*},a_{0},a_{1},n}]$, there exists some
sequence of reals $\\{c_{i,n}\\}_{i=1}^{n}$, such that the locally constant
potential

$\phi(x)=c_{x_{0},n}$

for $x=\cdots x_{-1}x_{0}x_{1}\cdots\in[x_{0}]$ on the full shift space
$\Lambda_{n}^{\mathbb{Z}}$ satisfies

(1.6)
$P(t\phi)=a_{0}+a_{1}(t-t_{*})+\cfrac{a_{2}}{2!}(t-t_{*})^{2}+O((t-t_{*})^{3})$

on $[t_{*}-\delta_{n},t_{*}+\delta_{n}]$ for some $\delta_{n}>0$.

This means we can fit some germs at $t_{*}$ up to level $2$ by pressures of
some locally constant potentials when the number of symbols of the shift space
is large enough. The values $\delta_{n},\\{c_{i,n}\\}_{i=1}^{n}$ all depend on
$t_{*},a_{0},a_{1},n$ and $a_{2}$ in fact, while we only indicate the
dependence of $m_{t_{*},a_{0},a_{1},n}$ and $M_{t_{*},a_{0},a_{1},n}$ as we
are particularly interested in their values in the context of Theorem 1.8.
There are some results on the values of

$\\{m_{t_{*},a_{0},a_{1},n},M_{t_{*},a_{0},a_{1},n}\\}_{n\in\mathbb{N}}$

subject to $t_{*}>0$ and $(a_{0},a_{1})\in\mathbb{R}^{2}$ satisfying (1.5) at
the end of Section 5.

We choose to present all our results in the one dimensional case, while many
of these results can in fact be extended to convex Lipschitz or analytic
functions $F(t_{1},t_{2},\cdots,t_{m})$ of $m$ variables naturally. Most of
our results also hold on transitive subshift spaces of finite type, with some
technical adjustments in their proofs involving the transition matrix.

The organization of the work is as following. In Section 2 we introduce some
basics in thermodynamic formalism and the Central Limit Theorem for the
process generated by a potential and the shift map on the symbolic space of
finite type. We give an explicit bound on the tail term in the CLT. Section 3
is devoted to the proof of Theorem 1.1 and 1.2. We formulate some expression
of the derivatives of the pressure (Corollary 3.11) linking directly to the
CLT, which allows us to unveil the relationship between derivatives of the
pressure function of various orders. Section 4 is devoted to the proof of
Theorem 1.7. In Section 5 we consider fitting 1- and 2-level candidate
analytic germs locally by pressure functions of locally constant potentials
(Problem 5.2) on symbolic spaces of finite type. We conjecture that any
reasonable analytic germ of finite level can be fitted by the pressure
function of some locally constant potential locally, as long as the number of
the symbols is large enough.

##  2\. Thermodynamic formalism and the CLT

In this section we collect some basic notions and results in thermodynamic
formalism for later use. We start from the pressure. Let $\Lambda$ be some
symbolic set of finite symbols, $\Lambda^{\mathbb{N}}$ be the shift space
equipped with the visual metric

$d(x,y)=\cfrac{1}{2^{l(x,y)}}$

for distinct
$x=x_{0}x_{1}x_{2}\cdots,y=y_{0}y_{1}y_{2}\cdots\in\Lambda^{\mathbb{N}}$, in
which

$l(x,y)=\min\\{i\in\mathbb{N}:x_{i}\neq y_{i}\\}$.

For a continuous potential $\phi:\Lambda^{\mathbb{N}}\rightarrow\mathbb{R}$ on
the compact metric space $\Lambda^{\mathbb{N}}$, Let

$S_{m,\phi}(x)=\sum_{i=0}^{m-1}\phi\circ\sigma^{i}(x)$

for $m\in\mathbb{N}$, in which $\sigma$ is the shift map.

######  2.1 Definition.

The pressure $P(\phi)$ of a continuous potential $\phi$ on
$\Lambda^{\mathbb{N}}$ is defined to be

$P(\phi)=\lim_{m\rightarrow\infty}\cfrac{1}{m}\log\sum_{\sigma^{m}(x)=x}e^{S_{m,\phi}(x)}$.

One can refer to [Wal1, p208] for a definition for continuous potentials on
general compact metric spaces. It satisfies the well-known variational formula

$P(\phi)=\sup\\{h(\mu)+\int\phi d\mu:\mu\mbox{ is a }\sigma-\mbox{invariant
measure on }\Lambda^{\mathbb{N}}\\}$.

Let $C^{0}(\Lambda^{\mathbb{N}})$ be the collection of all the continuous
potentials on $\Lambda^{\mathbb{N}}$. Two potentials $\psi,\phi\in
C^{0}(\Lambda^{\mathbb{N}})$ are said to be _cohomologous_ [Wal2] in case
there exists a continuous map
$\varphi:\Lambda^{\mathbb{N}}\rightarrow\mathbb{R}$ such that

$\psi(x)-\phi(x)=\varphi(x)-\varphi\circ\sigma(x)$.

We write $\psi\sim\phi$ to denote the equivalence relationship between two
potentials cohomologous to each other. The maps in

$\\{\varphi(x)-\varphi\circ\sigma(x):\varphi\in
C^{0}(\Lambda^{\mathbb{N}})\\}$

are called _coboundaries_. The importance of the cohomologous relationship is
revealed in the following result.

######  2.2 Proposition.

If $\psi\sim\phi$, then $P(\psi)=P(\phi)$. Moreover, $\psi$ and $\phi$ share
the same equilibrium state.

One can find a proof in [Rue1] or [PP]. Another important tool in
thermodynamic formalism is the _Ruelle operator_.

######  2.3 Definition.

For a continuous potential $\psi:\Lambda^{\mathbb{N}}\rightarrow\mathbb{R}$,
define the Ruelle operator $\mathcal{L}_{\psi}$ acting on
$C^{0}(\Lambda^{\mathbb{N}})$ as

$(\mathcal{L}_{\psi}f)(x)=\sum_{y:\sigma(y)=x}e^{\psi(y)}f(y)$

for $f\in C^{0}(\Lambda^{\mathbb{N}})$.

One can see easily that its compositions satisfy

(2.1)
$(\mathcal{L}^{m}_{\psi}f)(x)=\sum_{y:\sigma^{m}(y)=x}e^{S_{m,\psi}(y)}f(y)$

for any $m\in\mathbb{N}$. In case of $\psi$ being Hölder, it admits a simple
maximum isolated eigenvalue $\lambda=e^{P(\psi)}$ such that,

(2.2) $(\mathcal{L}_{\psi}w_{\psi})(x)=e^{P(\psi)}w_{\psi}(x)$

for some eigenfunction $w_{\psi}(x)\in C^{0,h}(\Lambda^{\mathbb{N}})$, refer
to [Rue1]. The unique equilibrium measure for the Hölder potential $\psi$ is
denoted by $\mu_{\psi}$. It then follows that

(2.3) $(\mathcal{L}^{m}_{\psi}w_{\psi})(x)=e^{mP(\psi)}w_{\psi}(x)$

for $w_{\psi}(x)\in C^{0,h}(\Lambda^{\mathbb{N}})$. A potential $\psi$ is said
to be _normalized_ if

$P(\psi)=0$ and $w_{\psi}=1_{\Lambda^{\mathbb{N}}}$,

in which $1_{\Lambda^{\mathbb{N}}}$ is the identity map on
$\Lambda^{\mathbb{N}}$. In case of $\psi$ being not normalized, we call

$\bar{\psi}=\psi+\log w_{\psi}-\log w_{\psi}\circ\sigma-P(\psi)$

the _normalization_ of $\psi$. It is easy to check that $\bar{\psi}$ is a
normalized potential. Moreover, $\bar{\psi}$ and $\psi$ share the same
equilibrium state.

Now we turn to the Central Limit Theorem for the random process
$\\{\phi\circ\sigma^{j}(x)\\}_{j=0}^{\infty}$ with the equilibrium measure
$\mu_{\psi}$ defined by some Hölder potential $\psi$, while $\phi$ is also
assumed to be Hölder. It deals with the asymptotic behaviour of the
distribution of $\cfrac{S_{m,\phi}}{\sqrt{m}}$ with respect to $\mu_{\psi}$ as
$m\rightarrow\infty$. The Ruelle operator comes in here, see [CP, Lal, Rou].
Let

$G_{m}(y)=\mu_{\psi}\Big{\\{}x\in\Lambda^{\mathbb{N}}:\cfrac{S_{m,\phi}(x)}{\sqrt{m}}<y\Big{\\}}$

for $y\in\mathbb{R}$. For $a,b\in\mathbb{R}$ and $b>0$, Let $N_{a,b}(y)$ be
the normal distribution with expectation $a$ and standard deviation $\sqrt{b}$
on $\mathbb{R}$, that is,

$\cfrac{dN_{a,b}(y)}{dy}=\cfrac{1}{\sqrt{2\pi b}}e^{-(y-a)^{2}/2b}$

for $y\in\mathbb{R}$. For Hölder potentials $\psi,\phi$ on a shift space of
finite type, since the pressure $P(\psi+t\phi)$ is analytic in a small
neighbourhood around $0$, denote by

$\Delta_{m}=P^{(m)}(\psi+t\phi)|_{t=0}$

for $m\in\mathbb{N}$ for convenience, while the readers can understand its
dependence on $\psi,\phi$ easily from the contexts in the following. Let

$P(\psi+t\phi)=\sum_{m=0}^{\infty}\cfrac{\Delta_{m}}{m!}t^{m}=\sum_{m=0}^{3}\cfrac{\Delta_{m}}{m!}t^{m}+t^{4}\kappa(t)$,

in which $\kappa(t)=\sum_{m=0}^{\infty}\cfrac{\Delta_{m+4}}{(m+4)!}t^{m}$.

######  Central Limit Theorem.

Let $\psi,\phi$ be Hölder potentials on a shift space of finite type with
$\phi$ being not cohomologous to a constant. If $\int\phi d\mu_{\psi}=0$, we
have

$\lim_{m\rightarrow\infty}G_{m}(y)=N_{0,\Delta_{2}}(y)+O(1/\sqrt{m})$,

in which

(2.4)
$O(1/\sqrt{m})\leq\cfrac{9|\Delta_{3}|+2|\Delta_{4}|}{\sqrt{2\pi^{3}m}(\Delta_{2})^{3/2}}.$

The convergence is uniform with respect to $y$. In case of $\phi$ being
generic, the result can be strengthened to

$\lim_{m\rightarrow\infty}G_{m}(y)=N_{0,\Delta_{2}}(y)+H_{m}(y)+o(1/\sqrt{m})$,

in which
$H_{m}(y)=\cfrac{\Delta_{3}}{6\sqrt{m}}\Big{(}1-\cfrac{y^{2}}{\Delta_{2}}\Big{)}e^{-\frac{y^{2}}{2\Delta_{2}}}$.

This fits into special cases of the _Berry-Esseen Theorem_ [Fel]. There is
nothing new in the version here comparing with [CP, Theorem 2, Theorem 3] or
[PP, Theorem 4.13], except the explicit bound on the tail term $O(1/\sqrt{m})$
in (2.4). In the following we justify this explicit bound. To do this, let

$\chi_{m}(z)=\displaystyle\int e^{iz\frac{S_{m,\phi}}{\sqrt{m}}}d\mu_{\psi}$

be the Fourier transformation of $G_{m}(y)$. Note that the Fourier
transformation of $N_{0,\Delta_{2}}(y)$ is $e^{-\frac{z^{2}\Delta_{2}}{2}}$.

######  2.4 Lemma.

Let $\psi,\phi$ be Hölder potentials on a shift space of finite type with
$\phi$ being not cohomologous to a constant. For $\epsilon>0$ small enough, we
have

(2.5)
$\cfrac{1}{2\pi}\int_{0}^{\epsilon\sqrt{m}}\cfrac{1}{z}\Big{|}\chi_{m}(z)-e^{-\frac{z^{2}\Delta_{2}}{2}}\Big{|}dz\leq\cfrac{\sqrt{2}|\Delta_{3}|}{12\sqrt{\pi
m}(\Delta_{2})^{3/2}}$

for any $m\in\mathbb{N}$ large enough.

###### Proof.

According to [PP, (4.6)], we have

$\displaystyle\int_{0}^{\epsilon\sqrt{m}}\cfrac{1}{z}\Big{|}\chi_{m}(z)-e^{-\frac{z^{2}\Delta_{2}}{2}}+\cfrac{iz^{3}\Delta_{3}}{6\sqrt{m}}e^{-\frac{z^{2}\Delta_{2}}{2}}\Big{|}dz=O(1/m)$

for $\epsilon>0$ small enough. So

(2.6)
$\cfrac{1}{2\pi}\displaystyle\int_{0}^{\epsilon\sqrt{m}}\cfrac{1}{z}\Big{|}\chi_{m}(z)-e^{-\frac{z^{2}\Delta_{2}}{2}}\Big{|}dz\leq
O(1/m)+\cfrac{|\Delta_{3}|}{12\pi\sqrt{m}}\displaystyle\int_{0}^{\epsilon\sqrt{m}}z^{2}e^{-\frac{z^{2}\Delta_{2}}{2}}dz.$

Considering

$\displaystyle\int_{-\infty}^{\infty}z^{2}e^{-\frac{z^{2}\Delta_{2}}{2}}dz=\cfrac{\sqrt{2\pi}}{(\Delta_{2})^{3/2}}$,

we obtain (2.5) from (2.6). ∎

Equipped with Lemma 2.4 we can justify the explicit bound on the tail term in
the Central Limit Theorem in (2.4).

Proof of the tail term in CLT:

###### Proof.

Without loss of generality, suppose $\psi$ is normalized and $\int\phi
d\mu_{\psi}=0$. It suffices for us to justify (2.4) considering [CP, Theorem
2, Theorem 3]. Similar to the proof of [CP, Theorem 2], apply [Fel, Lemma 2]
with the cumulative functions $G_{m}(y)$ and $N_{0,\Delta_{2}}(y)$ in our
case, one gets (c.f. [CP, (20)])

(2.7)
$|G_{m}(y)-N_{0,\Delta_{2}}(y)|\leq\cfrac{1}{2\pi}\int_{0}^{\epsilon\sqrt{m}}\cfrac{1}{z}\Big{|}\chi_{m}(z)-e^{-\frac{z^{2}\Delta_{2}}{2}}\Big{|}dz+\cfrac{24}{\epsilon\sqrt{2m\pi^{3}\Delta_{2}}}.$

Now let us take

$\cfrac{1}{\epsilon}=\cfrac{2}{\Delta_{2}}\big{(}\cfrac{|\Delta_{3}|}{6}+\cfrac{|\Delta_{4}|}{24}+\delta\big{)}$

for some small $\delta>0$, such that it satisfies (c.f. [CP, (10)])

$\cfrac{1}{\epsilon}>\max\Big{\\{}\cfrac{2}{\Delta_{2}}\Big{(}\cfrac{|\Delta_{3}|}{6}+t\kappa(t)\Big{)},\cfrac{2}{\Delta_{2}}\kappa(t)\Big{\\}}$

for any $|t|<\epsilon$ in (2.7). Considering (2.5), we have

(2.8) $\begin{array}[]{ll}&|G_{m}(y)-N_{0,\Delta_{2}}(y)|\vspace{3mm}\\\
\leq&\cfrac{\sqrt{2}|\Delta_{3}|}{12\sqrt{\pi
m}(\Delta_{2})^{3/2}}+\cfrac{24}{\sqrt{2m\pi^{3}\Delta_{2}}}\cfrac{2}{\Delta_{2}}\big{(}\cfrac{|\Delta_{3}|}{6}+\cfrac{|\Delta_{4}|}{24}+\delta\big{)}\vspace{3mm}\\\
=&\cfrac{\sqrt{2}|\Delta_{3}|}{12\sqrt{\pi
m}(\Delta_{2})^{3/2}}+\cfrac{8|\Delta_{3}|}{\sqrt{2\pi^{3}m}(\Delta_{2})^{3/2}}+\cfrac{2|\Delta_{4}|}{\sqrt{2\pi^{3}m}(\Delta_{2})^{3/2}}+\cfrac{48\delta}{\sqrt{2\pi^{3}m}(\Delta_{2})^{3/2}}\vspace{3mm}\\\
\leq&\cfrac{9|\Delta_{3}|}{\sqrt{2\pi^{3}m}(\Delta_{2})^{3/2}}+\cfrac{2|\Delta_{4}|}{\sqrt{2\pi^{3}m}(\Delta_{2})^{3/2}}+\cfrac{48\delta}{\sqrt{2\pi^{3}m}(\Delta_{2})^{3/2}}.\end{array}$

Finally, let $\delta\rightarrow 0$ in (2.8), we get (2.4).

∎

We will deal with the pressure function $P(\psi+t\phi)$ for $t\geq 0$ and
$\psi,\phi\in C^{0,h}(\Lambda^{\mathbb{N}})$ for some $0<h\leq 1$ in the
following sections. By [Rue2], $P(\psi+t\phi)$ depends analytically on $t$ in
case that $\psi,\phi$ are Hölder. We will often assume that

$\int\phi d\mu_{\psi}=0$

in the following when dealing with the higher differentials of $P(\psi+t\phi)$
because if $\int\phi d\mu_{\psi}=c\neq 0$, we have

$P(\psi+t(\phi-c))=P(\psi+t\phi)-ct$,

then

(2.9)
$\cfrac{d^{n}P(\psi+t(\phi-c))}{dt^{n}}=\cfrac{d^{n}P(\psi+t\phi)}{dt^{n}}$

for any $n\geq 2$ while $\int(\phi-c)d\mu_{\psi}=0$. We can also assume that
$\psi$ is normalized when dealing with the differentials of $P(\psi+t\phi)$.
If this is not the case we can simply change $\psi$ to its normalization
$\bar{\psi}$ while

(2.10)
$\cfrac{d^{n}P(\psi+t\phi)}{dt^{n}}=\cfrac{d^{n}P(\bar{\psi}+t\phi)}{dt^{n}}$

for $n\geq 1$ because

$P(\bar{\psi}+t\phi)=P(\psi+t\phi)-P(\psi)$

for any $t\in\mathbb{R}$.

##  3\. Derivatives of the pressures of Hölder potentials

In this section we formulate some explicit expressions for the derivatives of
the pressure $P(t\phi)=P(t)$ in terms of the derivatives of the eigenfunction
of $\mathcal{L}_{t\phi}$ for $\phi\in C^{0,h}(\Lambda^{\mathbb{N}})$ with
respect to $t$. We give basically two expressions of the derivatives, one of
which allows the introduction of the random stochastic process
$\\{\phi\circ\sigma^{j}(x)\\}_{j=0}^{m}$ for $m\in\mathbb{N}$. Upon the
expression we prove Theorem 1.1 and 1.2 in virtue of the CLT for the random
process $\\{\phi\circ\sigma^{j}(x)\\}_{j=0}^{\infty}$.

First we define some basics to deal with the higher derivatives of
compositional functions by the _Faà di Bruno’s formula_. For an integer
$j\in\mathbb{N}$, we say

$\tau=\tau_{1}\tau_{2}\cdots\tau_{q}$

with $q\in\mathbb{N}$ is a _partition_ of $j$ if the non-increasing sequence
of positive integers $j\geq\tau_{1}\geq\tau_{2}\geq\cdots\geq\tau_{q}\geq 1$
satisfies $\sum_{i=1}^{q}\tau_{i}=j$. Denote the collection of all the
possible partitions of $j$ by $\mathfrak{P}(j)$. For example, Table 1 lists
all the partitions in $\mathfrak{P}(5)$.

Table 1. Partitions of $5$ 5 | q=1
---|---
4,1 | q=2
3,2 | q=2
3,1,1 | q=3
2,2,1 | q=3
2,1,1,1 | q=4
1,1,1,1,1 | q=5

We sometimes simply write $\tau$ to denote the set
$\\{\tau_{1},\tau_{2},\cdots,\tau_{q}\\}$ for convenience in the following, so
$\\#\tau=q$. Now for $\tau$ being a partition of $j\geq 1$, let
$\\{B_{j}^{\tau}\\}$ be the number of different choices of dividing a set of
$j$ different elements into $\\#\tau=q$ sets of sizes
$\\{\tau_{i}\\}_{i=1}^{q}$ respectively (with no order on the sets of
partitions). Set $B_{0}^{0}=1$ for convenience. For example, consider the
cases $j=5$ and $\tau=3,1,1$, the number of different choices of dividing a
set of $5$ different elements into $q=3$ sets of sizes $3,1,1$ respectively is

$C_{5}^{3}=10=B_{5}^{3,1,1}$.

Table 2 lists all the numbers $\\{B_{5}^{\tau}\\}_{\tau\in\mathfrak{P}(5)}$.

Table 2. The coefficients $B_{5}^{\tau}$ $B_{5}^{5}=1$
---
$B_{5}^{4,1}=5$
$B_{5}^{3,2}=10$
$B_{5}^{3,1,1}=10$
$B_{5}^{2,2,1}=15$
$B_{5}^{2,1,1,1}=10$
$B_{5}^{1,1,1,1,1}=1$

For a smooth map $f:X\rightarrow Y$ between two metric spaces $X,Y$ and some
partition $\tau=\tau_{1},\tau_{2},\cdots,\tau_{q}\in\mathfrak{P}(j)$ with
$j\geq 1$, let

$f^{(\tau)}(x)=f^{(\tau_{1})}(x)f^{(\tau_{2})}(x)\cdots f^{(\tau_{q})}(x)$

be the product of the derivatives. For $j=0$ and $\tau=0\in\mathfrak{P}(0)$,
set $f^{(0)}(x)=1$. Then for two smooth functions $f:X\rightarrow Y$ and
$g:Y\rightarrow Z$ between metric spaces $X,Y,Z$, we have

(3.1) $\cfrac{d^{j}(g\circ
f(x))}{dx^{j}}=\sum_{\tau\in\mathfrak{P}(j)}B_{j}^{\tau}g^{(\\#\tau)}(f(x))f^{(\tau)}(x)$

in virtue of Faà di Bruno’s formula.

Now we turn to the higher differentials of the pressure function. We start by
considering some standard case, then extend the result to the general case.

######  3.1 Theorem.

Let $\psi,\phi\in C^{0,h}(\Lambda^{\mathbb{N}})$ with $\psi$ being normalized
for some finite symbolic set $\Lambda$. Assume $\int\phi d\mu_{\psi}=0$, in
which $\mu_{\psi}$ is the equilibrium state of $\psi$. Let $w(t,x)$ be the
eigenfunction of the maximum isolated eigenvalue $e^{P(\psi+t\phi)}$ of
$\mathcal{L}_{\psi+t\phi}$, which depends analytically on $t$ in a small
neighbourhood of $0$. Considering the differentials of the pressure function
$P(\psi+t\phi)$ at $t=0$, we have

(3.2)
$\begin{array}[]{ll}P^{(n)}(\psi+t\phi)|_{t=0}=&\sum_{j=1}^{n}C_{n}^{j}\int_{\Lambda^{\mathbb{N}}}(\phi(x))^{j}w^{(n-j)}(0,x)d\mu_{\psi}(x)\\\
&-\sum_{j=2}^{n-2}C_{n}^{j}\sum_{\tau\in\mathfrak{P}(j),1\notin\tau}B_{j}^{\tau}P^{(\tau)}(\psi+t\phi)|_{t=0}\int_{\Lambda^{\mathbb{N}}}w^{(n-j)}(0,x)d\mu_{\psi}(x)\\\
&-\sum_{\tau\in\mathfrak{P}(n),\\{1,n\\}\cap\tau=\emptyset}B_{n}^{\tau}P^{(\tau)}(\psi+t\phi)|_{t=0}\end{array}$

for any $n\geq 2$.

###### Proof.

According to the above notations, note that

(3.3) $(\mathcal{L}_{\psi+t\phi}w(t,\cdot))(x)=e^{P(\psi+t\phi)}w(t,x).$

The $n$-th derivative of
$(\mathcal{L}_{\psi+t\phi}w(t,\cdot))(x)=\sum_{y:\sigma(y)=x}e^{\psi(y)+t\phi(y)}w(t,y)$
gives

(3.4)
$\begin{array}[]{ll}&\cfrac{d^{n}\mathcal{L}_{\psi+t\phi}w(t,\cdot)(x)}{dt^{n}}\vspace{1mm}\\\
=&\sum_{y:\sigma(y)=x}\sum_{j=0}^{n}C_{n}^{j}\cfrac{d^{j}e^{(\psi+t\phi)(y)}}{dt^{j}}w^{(n-j)}(t,y)\vspace{3mm}\\\
=&\sum_{y:\sigma(y)=x}\sum_{j=0}^{n}C_{n}^{j}e^{(\psi+t\phi)(y)}(\phi(y))^{j}w^{(n-j)}(t,y)\vspace{3mm}\\\
=&\sum_{j=0}^{n}C_{n}^{j}\mathcal{L}_{\psi+t\phi}\big{(}(\phi(\cdot))^{j}w^{(n-j)}(t,\cdot)\big{)}.\end{array}$

All differentials are with respect to $t$. In case of $t=0$ this means

(3.5)
$\cfrac{d^{n}\mathcal{L}_{\psi+t\phi}w(t,\cdot)(x)}{dt^{n}}|_{t=0}=\sum_{j=0}^{n}C_{n}^{j}\mathcal{L}_{\psi}\big{(}(\phi(\cdot))^{j}w^{(n-j)}(0,\cdot)\big{)}.$

Note that the dual operator $\mathcal{L}_{\psi}^{*}$ fixes $\mu_{\psi}$, so
integration of both sides of (3.5) gives

(3.6)
$\int\cfrac{d^{n}\mathcal{L}_{\psi+t\phi}w(t,\cdot)(x)}{dt^{n}}|_{t=0}d\mu_{\psi}(x)=\sum_{j=0}^{n}C_{n}^{j}\int(\phi(x))^{j}w^{(n-j)}(0,x)\mu_{\psi}(x).$

In order to get the $n$-th derivative of $P(\psi+t\phi)$, differentiate
$e^{P(\psi+t\phi)}w(t,x)$ for $n$ times by (3.1), we get

(3.7)
$\begin{array}[]{ll}&\cfrac{d^{n}\Big{(}e^{P(\psi+t\phi)}w(t,x)\Big{)}}{dt^{n}}\vspace{1mm}\\\
=&\sum_{j=0}^{n}C_{n}^{j}\cfrac{d^{j}e^{P(\psi+t\phi)}}{dt^{j}}w^{(n-j)}(t,x)\vspace{3mm}\\\
=&\sum_{j=0}^{n-1}C_{n}^{j}\cfrac{d^{j}e^{P(\psi+t\phi)}}{dt^{j}}w^{(n-j)}(t,x)+\cfrac{d^{n}e^{P(\psi+t\phi)}}{dt^{n}}w(t,x)\vspace{3mm}\\\
=&\sum_{j=0}^{n-1}C_{n}^{j}\sum_{\tau\in\mathfrak{P}(j)}B_{j}^{\tau}P^{(\tau)}(\psi+t\phi)e^{P(\psi+t\phi)}w^{(n-j)}(t,x)\vspace{3mm}\\\
&+\sum_{\tau\in\mathfrak{P}(n)}B_{n}^{\tau}P^{(\tau)}(\psi+t\phi)e^{P(\psi+t\phi)}w(t,x)\vspace{3mm}\\\
=&\sum_{j=0}^{n-1}C_{n}^{j}\Big{(}\sum_{\tau\in\mathfrak{P}(j),1\notin\tau}B_{j}^{\tau}P^{(\tau)}(\psi+t\phi)+\sum_{\tau\in\mathfrak{P}(j),1\in\tau}B_{j}^{\tau}P^{(\tau)}(\psi+t\phi)\Big{)}e^{P(\psi+t\phi)}w^{(n-j)}(t,x)\vspace{3mm}\\\
&+\sum_{\tau\in\mathfrak{P}(n),n\notin\tau}B_{n}^{\tau}P^{(\tau)}(\psi+t\phi)e^{P(\psi+t\phi)}w(t,x)+P^{(n)}(\psi+t\phi)e^{P(\psi+t\phi)}w(t,x).\vspace{3mm}\\\
\end{array}$

Remember $P(\psi)=0$ and $w(0,x)=1$ as $\psi$ is normalized ([PP, p66]). Take
$t=0$ in (3.7) we get

(3.8)
$\begin{array}[]{ll}&\cfrac{d^{n}\Big{(}e^{P(\psi+t\phi)}w(t,x)\Big{)}}{dt^{n}}|_{t=0}\vspace{1mm}\\\
=&\sum_{j=0}^{n-1}C_{n}^{j}\Big{(}\sum_{\tau\in\mathfrak{P}(j),1\notin\tau}B_{j}^{\tau}P^{(\tau)}(\psi+t\phi)|_{t=0}+\sum_{\tau\in\mathfrak{P}(j),1\in\tau}B_{j}^{\tau}P^{(\tau)}(\psi+t\phi)|_{t=0}\Big{)}w^{(n-j)}(0,x)\vspace{3mm}\\\
&+\sum_{\tau\in\mathfrak{P}(n),n\notin\tau}B_{n}^{\tau}P^{(\tau)}(\psi+t\phi)|_{t=0}+P^{(n)}(\psi+t\phi)|_{t=0}\vspace{3mm}\\\
\end{array}$

Since $\int\phi d\mu_{\psi}=P^{\prime}(\psi+t\phi)|_{t=0}=0$ and $\int
w^{\prime}(0,x)d\mu_{\psi}=0$ ([PP, p66]), integrate both sides of (3.8) with
respect to $\mu_{\psi}$, we get

(3.9)
$\begin{array}[]{ll}&\displaystyle\int\cfrac{d^{n}\Big{(}e^{P(\psi+t\phi)}w(t,x)\Big{)}}{dt^{n}}|_{t=0}d\mu_{\psi}\vspace{1mm}\\\
=&\sum_{j=0}^{n-1}C_{n}^{j}\sum_{\tau\in\mathfrak{P}(j),1\notin\tau}B_{j}^{\tau}P^{(\tau)}(\psi+t\phi)|_{t=0}\int
w^{(n-j)}(0,x)d\mu_{\psi}\vspace{3mm}\\\
&+\sum_{\tau\in\mathfrak{P}(n),\\{1,n\\}\cap\tau=\emptyset}B_{n}^{\tau}P^{(\tau)}(\psi+t\phi)|_{t=0}+P^{(n)}(\psi+t\phi)|_{t=0}.\vspace{3mm}\\\
\end{array}$

Finally, combining (3.6) and (3.9) together we get (3.2).

∎

######  3.2 Remark.

The terms

$-\sum_{j=2}^{n-2}C_{n}^{j}\sum_{\tau\in\mathfrak{P}(j),1\notin\tau}B_{j}^{\tau}P^{(\tau)}(\psi+t\phi)|_{t=0}\int_{\Lambda^{\mathbb{N}}}w^{(n-j)}(0,x)d\mu_{\psi}(x)$

and

$-\sum_{\tau\in\mathfrak{P}(n),\\{1,n\\}\cap\tau=\emptyset}B_{n}^{\tau}P^{(\tau)}(\psi+t\phi)|_{t=0}$

in (3.2) are null in case of $n\leq 3$. This also applies to the corresponding
terms later.

######  3.3 Remark.

These appear to be inductive formulas, while one can always get non-inductive
ones via substituting the lower differentials $P^{(\tau)}(\psi+t\phi)|_{t=0}$
by their non-inductive versions depending only on
$\phi(x),\\{w^{(j)}(0,x)\\}_{j=1}^{n}$ and $\mu_{\psi}(x)$. This also applies
to Theorem 3.7.

One can find some description of derivatives of the pressure function by
_covariance_ of the sequence of functions
$\\{\phi\circ\sigma^{j}\\}_{j\in\mathbb{N}}$ in [KS1, Corollary 1] for smooth
$\phi$. Without the assumptions of $\psi$ being normalized and $\int\phi
d\mu_{\psi}=0$, Theorem 3.1 evolves into the following form.

######  3.4 Corollary.

Let $\psi,\phi\in C^{0,h}(\Lambda^{\mathbb{N}})$ with some finite symbolic set
$\Lambda$. $\mathcal{L}_{\psi+t\phi}$ admits a maximum isolated eigenvalue
$e^{P(\psi+t\phi)}$ close to $e^{P(\psi)}$ with eigenfunction $w(t,x)$ whose
projection depends analytically on $t$ in a small neighbourhood of $0$.
Considering the differentials of the pressure $P(\psi+t\phi)$ at $t=0$, we
have

(3.10)
$\begin{array}[]{ll}P^{(n)}(\psi+t\phi)|_{t=0}=&\sum_{j=1}^{n}C_{n}^{j}\displaystyle\int_{\Lambda^{\mathbb{N}}}\Big{(}\phi(x)-\int\phi
d\mu_{\psi}\Big{)}^{j}w^{(n-j)}(0,x)d\mu_{\psi}(x)\vspace{3mm}\\\
&-\sum_{j=2}^{n-2}C_{n}^{j}\sum_{\tau\in\mathfrak{P}(j),1\notin\tau}B_{j}^{\tau}P^{(\tau)}(\psi+t\phi)|_{t=0}\displaystyle\int_{\Lambda^{\mathbb{N}}}w^{(n-j)}(0,x)d\mu_{\psi}(x)\vspace{3mm}\\\
&-\sum_{\tau\in\mathfrak{P}(n),\\{1,n\\}\cap\tau=\emptyset}B_{n}^{\tau}P^{(\tau)}(\psi+t\phi)|_{t=0}\end{array}$

for any $n\geq 2$.

###### Proof.

Let

$\bar{\psi}=\psi+\log w_{\psi}(x)-\log w_{\psi}\circ\sigma-P(\psi)$

in which $w_{\psi}(x)$ is the eigenfunction of $\mathcal{L}_{\psi}$
corresponding to the eigenvalue $e^{P(\psi)}$. Take pressure in the following
equation

$\bar{\psi}+t\phi=\psi+t\phi+\log w_{\psi}(x)-\log
w_{\psi}\circ\sigma-P(\psi)$,

then apply Proposition 2.2, we see that

$P(\bar{\psi}+t\phi)=P(\psi+t\phi)-P(\psi)$.

This implies

(3.11)
$\cfrac{d^{n}P(\bar{\psi}+t\phi)}{dt^{n}}=\cfrac{d^{n}P(\psi+t\phi)}{dt^{n}}$

for any $n\geq 1$. Now apply Theorem 3.1 to the normalized potential
$\bar{\psi}$ and $\phi-\int\phi d\mu_{\psi}$, (note that
$\int\big{(}\phi-\int\phi d\mu_{\psi}\big{)}d\mu_{\psi}=0$ and
$\mu_{\psi}=\mu_{\bar{\psi}}$), we justify the corollary considering (3.11). ∎

In the following we present some concrete formulas of some special order $n$
in virtue of Theorem 3.1 for later use.

######  3.5 Corollary.

Let $\psi,\phi\in C^{0,h}(\Lambda^{\mathbb{N}})$ with $\psi$ being normalized.
Let $\mu_{\psi}$ be the equilibrium state of $\psi$ and $\int\phi
d\mu_{\psi}=0$. Let $e^{P(\psi+t\phi)}$ be the maximum eigenvalue of
$\mathcal{L}_{\psi+t\phi}$ with eigenfunction $w(t,x)$ for small $t$. Then we
have

(3.12) $P^{{}^{\prime\prime\prime}}(\psi+t\phi)|_{t=0}=3\int\phi
w^{{}^{\prime\prime}}(0,x)d\mu_{\psi}+3\int\phi^{2}w^{{}^{\prime}}(0,x)d\mu_{\psi}+\int\phi^{3}d\mu_{\psi}.$

###### Proof.

This follows instantly from Theorem 3.1 with $n=3$, along with some direct
computations on the Faà di Bruno’s coefficients
$\\{B_{3}^{\tau}\\}_{\tau\in\mathfrak{P}(3)}$. ∎

######  3.6 Corollary.

Let $\psi,\phi\in C^{0,h}(\Lambda^{\mathbb{N}})$ with $\psi$ being normalized.
Let $\mu_{\psi}$ be the equilibrium state of $\psi$ and $\int\phi
d\mu_{\psi}=0$. Let $e^{P(\psi+t\phi)}$ be the maximum eigenvalue of
$\mathcal{L}_{\psi+t\phi}$ with eigenfunction $w(t,x)$ for small $t$. Then we
have

(3.13)
$\begin{array}[]{ll}&P^{{}^{\prime\prime\prime\prime}}(\psi+t\phi)|_{t=0}\vspace{3mm}\\\
=&4\int\phi
w^{{}^{\prime\prime\prime}}(0,x)d\mu_{\psi}+6\int\phi^{2}w^{{}^{\prime\prime}}(0,x)d\mu_{\psi}+4\int\phi^{3}w^{{}^{\prime}}(0,x)d\mu_{\psi}+\int\phi^{4}d\mu_{\psi}\vspace{3mm}\\\
&-6P^{{}^{\prime\prime}}(\psi+t\phi)|_{t=0}\int
w^{{}^{\prime\prime}}(0,x)d\mu_{\psi}-3(P^{{}^{\prime\prime}}(\psi+t\phi)|_{t=0})^{2}\vspace{3mm}\\\
=&4\int\phi
w^{{}^{\prime\prime\prime}}(0,x)d\mu_{\psi}+6\int\phi^{2}w^{{}^{\prime\prime}}(0,x)d\mu_{\psi}+4\int\phi^{3}w^{{}^{\prime}}(0,x)d\mu_{\psi}+\int\phi^{4}d\mu_{\psi}\vspace{3mm}\\\
&-6(\int\phi^{2}d\mu_{\psi}+2\int\phi w^{\prime}(0,x)d\mu_{\psi})\int
w^{{}^{\prime\prime}}(0,x)d\mu_{\psi}-3(\int\phi^{2}d\mu_{\psi}+2\int\phi
w^{\prime}(0,x)d\mu_{\psi})^{2}.\end{array}$

###### Proof.

The first equality follows instantly from Theorem 3.1 with $n=4$ along with
some direct computations on the Faà di Bruno’s coefficients
$\\{B_{4}^{\tau}\\}_{\tau\in\mathfrak{P}(4)}$. The second one is true as

$P^{{}^{\prime\prime}}(\psi+t\phi)|_{t=0}=\int\phi^{2}d\mu_{\psi}+2\int\phi
w^{\prime}(0,x)d\mu_{\psi}.$

The latter description depends only on $\phi(x),\\{w^{(j)}(0,x)\\}_{j=1}^{3}$
and $\mu_{\psi}(x)$. ∎

One can also get some precise formulas for some particular $n$ in Corollary
3.4, and some non-inductive ones as we indicate in Remark 3.3. While the
formulas (3.2, 3.10, 3.12, 3.13) all give interesting descriptions of the
differentials of the pressure function $P(\psi+t\phi)$, it seems to us
difficult to discover any essential rigid restriction on them, or
relationships between them. In the following we turn to the description of
them by the random stochastic process
$\\{\phi\circ\sigma^{j}(x)\\}_{j=0}^{\infty}$. This is not a new idea on
exploring the regularity of the pressure function $P(\psi+t\phi)$, as one can
recall it from many others’ work in thermodynamic formalism. Again we first
consider some standard case, then extend to the general case.

######  3.7 Theorem.

Let $\psi,\phi\in C^{0,h}(\Lambda^{\mathbb{N}})$ with $\psi$ being normalized.
Let $\mu_{\psi}$ be the equilibrium state of $\psi$ and $\int\phi
d\mu_{\psi}=0$. Let $e^{P(\psi+t\phi)}$ be the maximum isolated eigenvalue of
$e^{P(\psi+t\phi)}$ with eigenfunction $w(t,x)$ whose projection depends
analytically on $t$. Considering the differentials of the pressure
$P(\psi+t\phi)$ at $t=0$, we have

(3.14) $\begin{array}[]{ll}&P^{(n)}(\psi+t\phi)|_{t=0}\\\
=&\lim_{m\rightarrow\infty}\cfrac{1}{m}\Big{(}\sum_{j=2}^{n}C_{n}^{j}\displaystyle\int_{\Lambda^{\mathbb{N}}}(S_{m,\phi}(x))^{j}w^{(n-j)}(0,x)d\mu_{\psi}(x)\vspace{3mm}\\\
&-\sum_{j=2}^{n-2}C_{n}^{j}\sum_{\tau\in\mathfrak{P}(j),1\notin\tau}m^{\\#\tau}B_{j}^{\tau}P^{(\tau)}(\psi+t\phi)|_{t=0}\displaystyle\int_{\Lambda^{\mathbb{N}}}w^{(n-j)}(0,x)d\mu_{\psi}(x)\vspace{3mm}\\\
&-\sum_{\tau\in\mathfrak{P}(n),\\{1,n\\}\cap\tau=\emptyset}m^{\\#\tau}B_{n}^{\tau}P^{(\tau)}(\psi+t\phi)|_{t=0}\Big{)}\end{array}$

for any $n\geq 2$.

###### Proof.

The proof follows the routine of Proof of Theorem 3.1. Considering (2.1), we
do $n$-differentials on both sides of (2.3), take $t=0$, then integrate both
sides with respect to $\mu_{\psi}(x)$, divided by $m$, we get

(3.15) $\begin{array}[]{ll}&P^{(n)}(\psi+t\phi)|_{t=0}\\\
=&\cfrac{1}{m}\Big{(}\sum_{j=1}^{n}C_{n}^{j}\displaystyle\int_{\Lambda^{\mathbb{N}}}(S_{m,\phi}(x))^{j}w^{(n-j)}(0,x)d\mu_{\psi}(x)\vspace{3mm}\\\
&-\sum_{j=2}^{n-2}C_{n}^{j}\sum_{\tau\in\mathfrak{P}(j),1\notin\tau}m^{\\#\tau}B_{j}^{\tau}P^{(\tau)}(\psi+t\phi)|_{t=0}\displaystyle\int_{\Lambda^{\mathbb{N}}}w^{(n-j)}(0,x)d\mu_{\psi}(x)\vspace{3mm}\\\
&-\sum_{\tau\in\mathfrak{P}(n),\\{1,n\\}\cap\tau=\emptyset}m^{\\#\tau}B_{n}^{\tau}P^{(\tau)}(\psi+t\phi)|_{t=0}\Big{)}\end{array}$

as (3.2). Now since $w^{(n-1)}(0,x)$ is bounded on $X$, the ergodic theorem
guarantees

(3.16)
$\lim_{m\rightarrow\infty}\cfrac{1}{m}\int_{\Lambda^{\mathbb{N}}}S_{m,\phi}(x)w^{(n-1)}(0,x)d\mu_{\psi}(x)=0.$

Then (3.14) follows from (3.15) as $m\rightarrow\infty$ considering (3.16). ∎

Theorem 3.7 establishes some link between the differentials of the pressure
function and the the process $\\{\phi\circ\sigma^{j}(x)\\}_{j=0}^{\infty}$
through $S_{m,\phi}$ with respect to the equilibrium state $\mu_{\psi}$. We
also formulate a general version of the result.

######  3.8 Corollary.

Let $\psi,\phi\in C^{0,h}(\Lambda^{\mathbb{N}})$ with $\mu_{\psi}$ be the
equilibrium state of $\psi$. $\mathcal{L}_{\psi+t\phi}$ admits a maximum
isolated eigenvalue $e^{P(\psi+t\phi)}$ close to $e^{P(\psi)}$ with
eigenfunction $w(t,x)$ whose projection depends analytically on $t$ in a small
neighbourhood of $0$. Considering the differentials of the pressure function
$P(\psi+t\phi)$ at $t=0$, we have

(3.17) $\begin{array}[]{ll}&P^{(n)}(\psi+t\phi)|_{t=0}\\\
=&\lim_{m\rightarrow\infty}\cfrac{1}{m}\Big{(}\sum_{j=2}^{n}C_{n}^{j}\displaystyle\int_{\Lambda^{\mathbb{N}}}\big{(}S_{m,\phi}-m\int\phi
d\mu_{\psi}\big{)}^{j}w^{(n-j)}(0,x)d\mu_{\psi}(x)\vspace{3mm}\\\
&-\sum_{j=2}^{n-2}C_{n}^{j}\sum_{\tau\in\mathfrak{P}(j),1\notin\tau}m^{\\#\tau}B_{j}^{\tau}P^{(\tau)}(\psi+t\phi)|_{t=0}\displaystyle\int_{\Lambda^{\mathbb{N}}}w^{(n-j)}(0,x)d\mu_{\psi}(x)\vspace{3mm}\\\
&-\sum_{\tau\in\mathfrak{P}(n),\\{1,n\\}\cap\tau=\emptyset}m^{\\#\tau}B_{n}^{\tau}P^{(\tau)}(\psi+t\phi)|_{t=0}\Big{)}\end{array}$

for any $n\geq 2$.

###### Proof.

Equipped with Theorem 3.7, the proof follows in line with the Proof of
Corollary 3.4. ∎

Now we give some precise descriptions of the third and fourth differentials of
$P(\psi+t\phi)$ in virtue of Theorem 3.7.

######  3.9 Corollary.

Let $\psi,\phi\in C^{0,h}(\Lambda^{\mathbb{N}})$ with $\psi$ being normalized.
Let $\mu_{\psi}$ be the equilibrium state of $\psi$ and $\int\phi
d\mu_{\psi}=0$. Let $e^{P(\psi+t\phi)}$ be the maximum eigenvalue of
$\mathcal{L}_{\psi+t\phi}$ with eigenfunction $w(t,x)$ for small $t$. Then we
have

(3.18)
$P^{{}^{\prime\prime}}(\psi+t\phi)|_{t=0}=\lim_{m\rightarrow\infty}\cfrac{1}{m}\int
S_{m,\phi}^{2}d\mu_{\psi}.$

######  3.10 Remark.

$P^{{}^{\prime\prime}}(\psi+t\phi)|_{t=0}$ is called variance of the random
process $\\{\phi\circ\sigma^{j}(x)\\}_{j=0}^{\infty}$, whose name can be
interpreted from the Central Limit Theorem. See [Rue1, PP].

######  3.11 Corollary.

Let $\psi,\phi\in C^{0,h}(\Lambda^{\mathbb{N}})$ with $\psi$ being normalized.
Let $\mu_{\psi}$ be the equilibrium state of $\psi$ and $\int\phi
d\mu_{\psi}=0$. Let $e^{P(\psi+t\phi)}$ be the maximum eigenvalue of
$\mathcal{L}_{\psi+t\phi}$ with eigenfunction $w(t,x)$ for small $t$. Then we
have

(3.19)
$P^{{}^{\prime\prime\prime}}(\psi+t\phi)|_{t=0}=\lim_{m\rightarrow\infty}\cfrac{3}{m}\int
S_{m,\phi}^{2}w^{{}^{\prime}}(0,x)d\mu_{\psi}+\lim_{m\rightarrow\infty}\cfrac{1}{m}\int
S_{m,\phi}^{3}d\mu_{\psi}.$

###### Proof.

This follows instantly from Theorem 3.7 with $n=3$. ∎

######  3.12 Corollary.

Let $\psi,\phi\in C^{0,h}(\Lambda^{\mathbb{N}})$ with $\psi$ being normalized.
Let $\mu_{\psi}$ be the equilibrium state of $\psi$ and $\int\phi
d\mu_{\psi}=0$. Let $e^{P(\psi+t\phi)}$ be the maximum eigenvalue of
$\mathcal{L}_{\psi+t\phi}$ with eigenfunction $w(t,x)$ for small $t$. Then we
have

(3.20) $\begin{array}[]{ll}&P^{(4)}(\psi+t\phi)|_{t=0}\vspace{3mm}\\\
=&\lim_{m\rightarrow\infty}\Big{(}\cfrac{6}{m}\displaystyle\int
S_{m,\phi}^{2}w^{{}^{\prime\prime}}(0,x)d\mu_{\psi}+\cfrac{4}{m}\int
S_{m,\phi}^{3}w^{{}^{\prime}}(0,x)d\mu_{\psi}+\cfrac{1}{m}\int
S_{m,\phi}^{4}d\mu_{\psi}\vspace{3mm}\\\
&-6P^{{}^{\prime\prime}}(\psi+t\phi)|_{t=0}\displaystyle\int
w^{{}^{\prime\prime}}(0,x)d\mu_{\psi}-3m(P^{{}^{\prime\prime}}(\psi+t\phi)|_{t=0})^{2}\Big{)}\vspace{3mm}\\\
=&\lim_{m\rightarrow\infty}\Big{(}\cfrac{6}{m}\displaystyle\int
S_{m,\phi}^{2}w^{{}^{\prime\prime}}(0,x)d\mu_{\psi}+\cfrac{4}{m}\int
S_{m,\phi}^{3}w^{{}^{\prime}}(0,x)d\mu_{\psi}+\cfrac{1}{m}\int
S_{m,\phi}^{4}d\mu_{\psi}\vspace{3mm}\\\ &-\cfrac{6}{m}\displaystyle\int
S_{m,\phi}^{2}d\mu_{\psi}\displaystyle\int
w^{{}^{\prime\prime}}(0,x)d\mu_{\psi}-\cfrac{3}{m}(\int
S_{m,\phi}^{2}d\mu_{\psi})^{2}\Big{)}\vspace{3mm}\\\ .\end{array}$

###### Proof.

The first equality follows instantly from Theorem 3.7 with $n=4$, while the
second one is true considering (3.18). The last description depends only on
$\phi(x),\\{w^{(j)}(0,x)\\}_{j=1}^{2}$ and $\mu_{\psi}(x)$. ∎

Through the above formulas we see the importance of the asymptotic
distribution of the random variable $S_{m,\phi}$ with respect to $\mu_{\psi}$,
which is describe by the Central Limit Theorem for the process
$\\{\phi\circ\sigma^{j}(x)\\}_{j=0}^{\infty}$. Equipped with all the above
results, now we are in a position to prove the rigidity results on the third
differentials of $P(\psi+t\phi)$ upon Corollary 3.11. We first show Theorem
1.2.

###### Proof of Theorem 1.2.

From now on we fix $t_{*}\in(0,\infty)$. Let $\psi=t_{*}\phi$. Simply by
making a change of variable we can see that

$P^{(n)}(t_{*})=P^{(n)}(t\phi)|_{t=t_{*}}=P^{(n)}(\psi+t\phi)|_{t=0}$

for any $n\geq 0$. So (1.2) is equivalent to

(3.21)
$|P^{\prime\prime\prime}(\psi+t\phi)|_{t=0}\big{(}1-\sqrt{2\pi}(P^{\prime\prime}(\psi+t\phi)|_{t=0})^{3/2}\big{)}|\leq
3M_{\phi}P^{\prime\prime}(\psi+t\phi)|_{t=0}.$

We can assume $\psi$ is normalized as otherwise we can change it to its
normalization considering (2.10). Moreover, it suffices for us to prove it
under the assumption $\int\phi d\mu_{\psi}=0$ in virtue of (2.9). If
$P^{\prime\prime}(\psi+t\phi)|_{t=0}=0$, then $\phi$ is cohomologous to a
constant according to [PP, Proposition 4.12]. This forces
$P^{\prime\prime\prime}(\psi+t\phi)|_{t=0}=0$, so (3.21) is satisfied in this
case. In the following we assume $P^{\prime\prime}(\psi+t\phi)|_{t=0}>0$. We
resort to Corollary 3.11 to justify (3.21) under the above assumptions. We
first estimate the term $\cfrac{1}{m}\int S_{m,\phi}^{3}d\mu_{\psi}$ in
(3.19). Now the Central Limit Theorem comes in.

$\begin{array}[]{ll}&\cfrac{1}{m}\displaystyle\int
S_{m,\phi}^{3}d\mu_{\psi}\vspace{3mm}\\\
=&\sqrt{m}\displaystyle\int\Big{(}\cfrac{S_{m,\phi}}{\sqrt{m}}\Big{)}^{3}d\mu_{\psi}\vspace{3mm}\\\
=&\sqrt{m}\displaystyle\int y^{3}dG_{m}(y)\vspace{3mm}\\\
=&\sqrt{m}\displaystyle\int
y^{3}dN_{0,P^{\prime\prime}(\psi+t\phi)|_{t=0}}(y)+\sqrt{m}\int
y^{3}dH_{m}(y)+\sqrt{m}\cdot o(1/\sqrt{m})\vspace{3mm}\\\ =&\sqrt{m}\cdot
0+\displaystyle\int
y^{3}d\Big{(}\cfrac{P^{\prime\prime\prime}(\psi+t\phi)|_{t=0}}{6}\big{(}1-\cfrac{y^{2}}{P^{\prime\prime}(\psi+t\phi)|_{t=0}}\big{)}e^{-y^{2}/2P^{\prime\prime}(\psi+t\phi)|_{t=0}}\Big{)}\\\
&+\sqrt{m}\cdot o(1/\sqrt{m})\vspace{3mm}\\\
=&P^{\prime\prime\prime}(\psi+t\phi)|_{t=0}\sqrt{2\pi}(P^{\prime\prime}(\psi+t\phi)|_{t=0})^{3/2}+\sqrt{m}\cdot
o(1/\sqrt{m}).\vspace{3mm}\\\ \end{array}$

By taking $m\rightarrow\infty$ we get

(3.22) $\lim_{m\rightarrow\infty}\cfrac{1}{m}\displaystyle\int
S_{m,\phi}^{3}d\mu_{\psi}=P^{\prime\prime\prime}(\psi+t\phi)|_{t=0}\sqrt{2\pi}(P^{\prime\prime}(\psi+t\phi)|_{t=0})^{3/2}.$

Considering (3.19) we have

(3.23)
$P^{\prime\prime\prime}(\psi+t\phi)|_{t=0}\big{(}1-\sqrt{2\pi}(P^{\prime\prime}(\psi+t\phi)|_{t=0})^{3/2}\big{)}=\lim_{m\rightarrow\infty}\cfrac{3}{m}\int
S_{m,\phi}^{2}w^{{}^{\prime}}(0,x)d\mu_{\psi}.$

Since $w^{{}^{\prime}}(0,x)$ depends continuously on $x\in X$, there exists
some $M_{\phi}$ depending on $\phi$, such that

(3.24) $|w^{{}^{\prime}}(0,x)|\leq M_{\phi}.$

Now taking absolute values on both sides of (3.23) we justify (3.21),
considering (3.24) and (3.18).

∎

The proof of Theorem 1.1 on the pressure functions of non-generic Hölder
potentials follows a similar way.

###### Proof of Theorem 1.1.

Fix $t_{*}\in(0,\infty)$, we can simply assume $\psi=t_{*}\phi$ is normalised
and $\int\phi d\mu_{\psi}=0$. In case that
$P^{\prime\prime}(\psi+t\phi)|_{t=0}=0$, so $\phi$ is cohomologous to a
constant, (1.1) holds obviously. In the following we assume $\phi$ is not
cohomologous to a constant, so $P^{\prime\prime}(\psi+t\phi)|_{t=0}>0$. We
again resort to Corollary 3.11 to justify (1.1) under these assumptions. Now
for the term $\cfrac{1}{m}\int S_{m,\phi}^{3}d\mu_{\psi}$ in (3.19), in virtue
of the Central Limit Theorem,

(3.25) $\begin{array}[]{ll}&\cfrac{1}{m}\displaystyle\int
S_{m,\phi}^{3}d\mu_{\psi}\vspace{3mm}\\\
=&\sqrt{m}\displaystyle\int\Big{(}\cfrac{S_{m,\phi}}{\sqrt{m}}\Big{)}^{3}d\mu_{\psi}\vspace{3mm}\\\
=&\sqrt{m}\displaystyle\int y^{3}dG_{m}(y)\vspace{3mm}\\\
\leq&\sqrt{m}\displaystyle\int
y^{3}dN_{0,P^{\prime\prime}(\psi+t\phi)|_{t=0}}(y)+\sqrt{m}\cfrac{9|P^{\prime\prime\prime}(\psi+t\phi)|_{t=0}|+2|P^{(4)}(\psi+t\phi)|_{t=0}|}{\sqrt{2\pi^{3}m}(P^{\prime\prime}(\psi+t\phi)|_{t=0})^{3/2}}\vspace{3mm}\\\
=&\sqrt{m}\cdot
0+\cfrac{9|P^{\prime\prime\prime}(\psi+t\phi)|_{t=0}|+2|P^{(4)}(\psi+t\phi)|_{t=0}|}{\sqrt{2\pi^{3}}(P^{\prime\prime}(\psi+t\phi)|_{t=0})^{3/2}}\vspace{3mm}\\\
=&\cfrac{9|P^{\prime\prime\prime}(\psi+t\phi)|_{t=0}|+2|P^{(4)}(\psi+t\phi)|_{t=0}|}{\sqrt{2\pi^{3}}(P^{\prime\prime}(\psi+t\phi)|_{t=0})^{3/2}}\\\
\end{array}$

for $m$ large enough. By taking $m\rightarrow\infty$ in (3.25), we get

(3.26) $\lim_{m\rightarrow\infty}\cfrac{1}{m}\displaystyle\int
S_{m,\phi}^{3}d\mu_{\psi}\leq\cfrac{9|P^{\prime\prime\prime}(\psi+t\phi)|_{t=0}|+2|P^{(4)}(\psi+t\phi)|_{t=0}|}{\sqrt{2\pi^{3}}(P^{\prime\prime}(\psi+t\phi)|_{t=0})^{3/2}}.$

Taking modulus on both sides of (3.26) we get

(3.27)
$\begin{array}[]{ll}&|P^{\prime\prime\prime}(\psi+t\phi)|_{t=0}|\vspace{3mm}\\\
\leq&\Big{|}\lim_{m\rightarrow\infty}\cfrac{1}{m}\displaystyle\int
S_{m,\phi}^{3}d\mu_{\psi}\Big{|}+\Big{|}\lim_{m\rightarrow\infty}\cfrac{3}{m}\int
S_{m,\phi}^{2}w^{{}^{\prime}}(0,x)d\mu_{\psi}\Big{|}\vspace{3mm}\\\
\leq&\cfrac{9|P^{\prime\prime\prime}(\psi+t\phi)|_{t=0}|+2|P^{(4)}(\psi+t\phi)|_{t=0}|}{\sqrt{2\pi^{3}}(P^{\prime\prime}(\psi+t\phi)|_{t=0})^{3/2}}+3M_{\phi}\Big{|}\lim_{m\rightarrow\infty}\cfrac{3}{m}\displaystyle\int
S_{m,\phi}^{2}d\mu_{\psi}\Big{|}\vspace{3mm}\\\
=&\cfrac{9|P^{\prime\prime\prime}(\psi+t\phi)|_{t=0}|+2|P^{(4)}(\psi+t\phi)|_{t=0}|}{\sqrt{2\pi^{3}}(P^{\prime\prime}(\psi+t\phi)|_{t=0})^{3/2}}+3M_{\phi}P^{\prime\prime}(\psi+t\phi)|_{t=0}\end{array}$

for some $|w^{{}^{\prime}}(0,x)|\leq M_{\phi}$, which results in (1.1). ∎

One can predict from Corollary 3.12, Theorem 3.7 and the proof of Theorem 1.1,
Theorem 1.2 that some more rigid relationships between higher differentials of
the pressure function $\\{P^{(n)}(t\phi)\\}_{n\in\mathbb{N}}$ are possible.
These rigidity relationships impose restrictions on fitting convex analytic
functions whose supporting lines intersecting the vertical axis in some
bounded set in $[0,\infty)$ by pressures of Hölder potentials.

##  4\. Global Fitting of convex analytic functions via pressures of Hölder
potentials

This section is dedicated to the proof of Theorem 1.7. We start from the
following result on some global behaviour of the pressure functions of generic
Hölder potentials.

######  4.1 Theorem.

Let $\alpha>0$. If a strictly convex analytic function $F(t)$ on
$(\alpha,\infty)$, with its supporting lines intersecting the vertical axis in
$[\underline{\gamma},\overline{\gamma}]\subset[0,\infty)$, such that

(4.1)
$\sup_{t\in(\alpha,\infty)}\Bigg{\\{}\Big{|}\cfrac{F^{{}^{\prime\prime\prime}}(t)}{F^{{}^{\prime\prime}}(t)}-\sqrt{2\pi
F^{{}^{\prime\prime}}(t)}\Big{|}\Bigg{\\}}=\infty,$

then there does not exist any generic Hölder potential $\phi$ on any shift
space $X$ of finite type satisfying

$P(t\phi)=F(t)$

on $(\alpha,\infty)$.

###### Proof.

This follows directly from Theorem 1.2 in fact. Suppose on the contrary that
there exist some shift space $X$ of finite type and some generic Hölder
potential $\phi\in C^{0,h}(X)$ satisfying $P(t\phi)=F(t)$ on
$(\alpha,\infty)$, then according to Theorem 1.2, we have

$\sup_{t\in(\alpha,\infty)}\Bigg{\\{}\Big{|}\cfrac{F^{{}^{\prime\prime\prime}}(t)}{F^{{}^{\prime\prime}}(t)}-\sqrt{2\pi
F^{{}^{\prime\prime}}(t)}\Big{|}\Bigg{\\}}\leq 3M_{\phi}$

for some finite $M_{\phi}>0$. This contradicts (4.1). ∎

Be careful that we cannot exclude the possibility that one can locally fit
some convex analytic function through the pressure of some generic Hölder
potential on some shift space of finite type by Theorem 1.2. This is because
for any strictly convex analytic function $F(t)$ on $(\alpha,\infty)$ and
$\alpha\leq\underline{\alpha}\leq\overline{\alpha}$, we always have

$\sup_{\underline{\alpha}\leq
t\leq\overline{\alpha}}\Bigg{\\{}\Big{|}\cfrac{F^{{}^{\prime\prime\prime}}(t)}{F^{{}^{\prime\prime}}(t)}-\sqrt{2\pi
F^{{}^{\prime\prime}}(t)}\Big{|}\Bigg{\\}}<\infty$.

So one cannot exclude the possibility that there exists some generic Hölder
potential $\phi$ on some shift space of finite type satisfying

$P(t\phi)=F(t)$

on $[\underline{\alpha},\overline{\alpha}]$ through Theorem 1.2. See Section 5
for more results on the problem of local fitting of some convex analytic
functions through the pressures of Hölder potentials.

Now for $\alpha>0$, let

$\begin{array}[]{ll}\mathcal{F}_{\alpha}=&\\{F(t):F(t)\mbox{ is a strictly
convex analytic function on }(\alpha,\infty)\mbox{ satisfying
}(\ref{eq26}),\vspace{3mm}\\\ &\mbox{ its supporting lines intersect the
vertical axis in a bounded interval in }[0,\infty)\\}.\vspace{3mm}\end{array}$

We will show that $\mathcal{F}_{\alpha}\neq\emptyset$ for any $\alpha>0$ in
the following.

######  4.2 Proposition.

For any $\alpha>0$, we have

$\tilde{\mathcal{F}}_{\alpha}=\Big{\\{}F_{a,b,c}(t)=\cfrac{at^{2}+bt+te^{-ct^{2}}+e^{-ct^{2}}}{t}\Big{|}_{(\alpha,\infty)}\Big{\\}}_{a,b>0,c>1/2\sqrt{2}}\subset\mathcal{F}_{\alpha}$.

###### Proof.

The restricted functions on $(\alpha,\infty)$ are of course analytic.
Considering the second derivative of a function
$F_{a,b,c}(t)\in\tilde{\mathcal{F}}_{\alpha}$, we have

$F_{a,b,c}^{{}^{\prime\prime}}(t)=4c^{2}t^{2}e^{-ct^{2}}+4c^{2}te^{-ct^{2}}-2ce^{-ct^{2}}+2ct^{-1}e^{-ct^{2}}+2t^{-3}e^{-ct^{2}}$

for $t\in(0,\infty)$. Now since

$4c^{2}t+2ct^{-1}\geq 2\sqrt{8c^{3}}>2c$

considering $c>1/2\sqrt{2}$, we can see that
$F_{a,b,c}^{{}^{\prime\prime}}(t)>0$ on $(0,\infty)$. This shows that for any
$\alpha>0$, $F_{a,b,c}(t)\in\tilde{\mathcal{F}}_{\alpha}$ is a convex
function. Considering the third differential of a function
$F_{a,b,c}(t)\in\tilde{\mathcal{F}}_{\alpha}$, we have

$F_{a,b,c}^{{}^{\prime\prime\prime}}(t)=-8c^{3}t^{3}e^{-ct^{2}}-8c^{3}t^{2}e^{-ct^{2}}+12c^{2}te^{-ct^{2}}-6ct^{-2}e^{-ct^{2}}-6t^{-4}e^{-ct^{2}}$

for $t\in(0,\infty)$. Then we have

$\lim_{t\rightarrow\infty}\Bigg{(}\cfrac{F^{{}^{\prime\prime\prime}}_{a,b,c}(t)}{F^{{}^{\prime\prime}}_{a,b,c}(t)}-\sqrt{2\pi
F^{{}^{\prime\prime}}_{a,b,c}(t)}\Bigg{)}=\lim_{t\rightarrow\infty}\cfrac{-8c^{3}t^{3}e^{-ct^{2}}}{4c^{2}t^{2}e^{-ct^{2}}}=-\infty$.

This means that $F_{a,b,c}(t)\in\tilde{\mathcal{F}}_{\alpha}$ satisfies (4.1).
To see that the supporting lines of a function
$F_{a,b,c}(t)\in\tilde{\mathcal{F}}_{\alpha}$ intersect the vertical axis in a
bounded domain in $[0,\infty)$, write the function as

$F_{a,b,c}(t)=at+b+e^{-ct^{2}}+t^{-1}e^{-ct^{2}}$.

Its graph on $(0,\infty)$ is a strictly convex smooth curve with asymptotes
$y=at+b$ and $t=0$. ∎

In Figure 1 we provide the readers with the graph of the function

$F_{2,3,1}(t)=\cfrac{2t^{2}+3t+te^{-t^{2}}+e^{-t^{2}}}{t}$

on $(0,\infty)$.

Figure 1.  Graph of $F_{2,3,1}(t)$

This means that any function in the family $\tilde{\mathcal{F}}_{\alpha}$
cannot be fitted by any generic Hölder potential on any shift space of finite
type globally, considering Theorem 4.1. In the following we deny the
possibility that they can be fitted by _non-generic_ Hölder potentials on
shift spaces of finite type.

######  4.3 Definition.

A continuous potential $\phi:X\to\mathbb{R}$ on a shift space $X$ of finite
type is said to be non-generic if for some normalised potential $\psi$, the
spectral radius of the complex Ruelle operator $\mathcal{L}_{\psi+it\phi}$
equals $1$ for some $t\neq 0$.

One can show that if $\phi$ is non-generic then there exists a continuous
function $u:X\to\mathbb{R}$, $c_{\phi}\in\mathbb{R}$ and a locally constant
potential $\tilde{\phi}:X\to\mathbb{R}$, such that

(4.2) $\phi(x)=u\circ\sigma(x)-u(x)+c_{\phi}+\tilde{\phi}(x).$

######  4.4 Proposition.

For any $\alpha>0$ and any $F_{a,b,c}(t)\in\tilde{\mathcal{F}}_{\alpha}$ with
$a,b>0,c>\frac{1}{2\sqrt{2}}$, there does not exist any non-generic Hölder
potential $\phi$ on any shift space of finite type such that

$P(t\phi)=F(t)$

on $(\alpha,\infty)$.

###### Proof.

Note that for a non-generic Hölder potential $\phi$ on a shift space of finite
type, according to (4.2), we have

$P(t\phi)=tc_{\phi}+P(t\tilde{\phi})$,

in which $\tilde{\phi}$ is some locally constant potential. By the explicit
formula (see Lemma 5.3) for the pressure functions of locally constant
potentials on shift spaces of finite type, we see that any $F_{a,b,c}(t)$
cannot be fitted by pressure of any non-generic Hölder potential $\phi$
globally. ∎

Equipped with all the above results, Theorem 1.7 follows instantly from
Proposition 4.2 and 4.4.

##  5\. Local fitting of analytic germs via pressures of locally constant
potentials

In this section we deal with the local fitting of analytic functions by the
pressures of Hölder potentials, especially the pressures of piecewise constant
ones. Firstly we borrow some notion originating from analytic continuation.

######  5.1 Definition.

A germ at $t_{*}$ is the sum of infinite power series

$g(t)=a_{0}+a_{1}(t-t_{*})+\cfrac{a_{2}}{2!}(t-t_{*})^{2}+\cfrac{a_{3}}{3!}(t-t_{*})^{3}+\cdots$

for some $(a_{0},a_{1},\cdots)\in\mathbb{R}^{\infty}$.

The convergent radius (the superior of values $\delta\geq 0$ on
$[t_{*}-\delta,t_{*}+\delta]$ the germ converges) of the power series is
called the _radius_ of the germ. We are only interested in germs of radius
$\delta>0$. The following problem will be our concern in this section.

######  5.2 Problem.

For a germ

$g(t)=a_{0}+a_{1}(t-t_{*})+\cfrac{a_{2}}{2!}(t-t_{*})^{2}+\cdots$

at $t_{*}$ with some strictly positive radius, does there exist some Hölder
potential $\phi$ on some shift space of finite type and some $\delta>0$, such
that

$P(t\phi)=g(t)$

on $[t_{*}-\delta,t_{*}+\delta]$?

The question can still be understood in Katok’s flexibility program in the
class of symbolic dynamical systems, or even in some smooth systems. Obvious
conditions to guarantee a positive answer to the problem are (1.5) and

(5.1) $a_{2}>0.$

Condition (5.1) guarantees convexity of the germ (in some neighbourhood of
$t_{*}$) while (1.5) guarantees the supporting lines of the germ intersect the
vertical axis in a bounded set in $[0,\infty)$ (also in some neighbourhood of
$t_{*}$). We are especially interested in its answer when the Hölder potential
in Problem 5.2 is required to be a piecewise constant one. We have seen the
importance of the family of locally constant potentials in approximating
convex analytic functions in Corollary 1.4. In fact Corollary 1.4 has some
interesting interpretation in approximation theory [Tim], when we consider the
explicit expressions of the pressures of locally constant potentials on the
shift space of finite type. For $n\in\mathbb{N}$, recall that

$\Lambda_{n}=\\{1,2,\cdots,n\\}$.

######  5.3 Lemma.

For an integer $k\geq 0$, consider some locally constant potential

$\phi(x)=c_{x_{-k}x_{-k+1}\cdots x_{0}\cdots x_{k-1}x_{k}}$

for $x=\cdots x_{-1}x_{0}x_{1}\cdots\in[x_{-k}\cdots x_{k}]$ on the shift
space $\Lambda_{n}^{\mathbb{Z}}$, we have

$P(t\phi)=\log\sum_{(x_{-k},\cdots,x_{k})\in\Lambda_{n}^{2k+1}}e^{tc_{x_{-k}\cdots
x_{k}}}$

for any $t\in(-\infty,\infty)$.

###### Proof.

This follows from [Wal1, Theorem 9.6] by some direct calculations through
Definition 2.1 of the pressure. See also [Wal1, p214]. ∎

######  5.4 Remark.

The result can be extended to transitive subshifts of finite type. In this
case the pressure is the logarithm of the maximal eigenvalue of some
appropriate matrix.

Now combining Corollary 1.4 and Lemma 5.3, we have the following result.

######  5.5 Corollary.

Let $F(t)$ be a convex Lipschitz function on $(\alpha,\infty)$ for some
$\alpha>0$, such that its supporting lines intersect the vertical axis in
$[\underline{\gamma},\overline{\gamma}]$ with
$0\leq\underline{\gamma}\leq\overline{\gamma}<\infty$. Then there exists some
$K\in\mathbb{N}$ and some sequences of constants

$\\{c_{n,j}\\}_{j=1}^{K^{n}}$,

such that

(5.2) $\lim_{n\rightarrow\infty}\log\sum_{j=1}^{K^{n}}e^{tc_{n,j}}=F(t)$

for any $t\in(\alpha,\infty)$.

###### Proof.

Take $K=\\#\Lambda$ for the symbolic set in the proof of Corollary 1.4, then
the locally constant potential $\phi_{n}(x)=\phi_{n,-}(x)$ admits $K^{n}$
constant values respectively on corresponding level-$n$ cylinder sets. Denote
these values by $\\{c_{n,j}\\}_{j=1}^{K^{n}}$ for $n\in\mathbb{N}$. According
to Lemma 5.3,

$P(t\phi_{n,-})=\log\sum_{j=1}^{K^{n}}e^{tc_{n,j}}$

for any $n\geq 1$. This gives (5.2) by virtue of (1.3).

∎

Corollary 5.5 indicates that logarithm of the finite sums of the exponential
maps in the family $\\{e^{tc}\\}_{c\in\mathbb{R}}$ are dense in the space of
certain convex Lipschitz maps on $(\alpha,\infty)$. The above approximation is
uniform with respect to $t$ in a bounded set. This makes the family
$\\{e^{tc}\\}_{c\in\mathbb{R}}$ (family of locally constant potentials)
important in detecting the properties of certain convex Lipschitz maps (among
continuous or Hölder potentials).

From now on we turn our attention to Problem 5.2, but with restriction on
locally constant potentials. We focus on locally constant potentials defined
on the level-$0$ cylinder sets, whose theory is presumably parallel to the
ones defined on the deeper cylinder sets. On the shift space
$\Lambda_{n}^{\mathbb{Z}}$ with $n\geq 2$, consider the locally constant
potential

$\phi(x)=z_{x_{0}}$

for $x=\cdots x_{-1}x_{0}x_{1}\cdots\in[x_{0}]$, in which $\\{z_{i}\\}_{1\leq
i\leq n}$ are all constants. Let

$Q_{0}(t,z_{1},z_{2},\cdots,z_{n})=\sum_{i=1}^{n}e^{tz_{i}}$,

so

$P(t\phi)=\log Q_{0}(t,z_{1},\cdots,z_{n})$

by Lemma 5.3. Let

$Q_{1}(t,z_{1},z_{2},\cdots,z_{n})=\sum_{i=1}^{n}z_{i}e^{tz_{i}}$

and

$Q_{2}(t,z_{1},z_{2},\cdots,z_{n})=\sum_{1\leq i<j\leq
n}(z_{i}-z_{j})^{2}e^{t(z_{i}+z_{j})}$.

Through some elementary calculations one can check that

$P^{\prime}(t\phi)=\cfrac{dP(t\phi)}{dt}=\cfrac{Q_{1}(t,z_{1},\cdots,z_{n})}{Q_{0}(t,z_{1},\cdots,z_{n})}$

while

(5.3)
$P^{\prime\prime}(t\phi)=\cfrac{d^{2}P(t\phi)}{dt^{2}}=\cfrac{Q_{2}(t,z_{1},\cdots,z_{n})}{Q_{0}^{2}(t,z_{1},\cdots,z_{n})}.$

Let

$R_{2}(t,z_{1},z_{2},\cdots,z_{n})=\sum_{i=1}^{n}z_{i}^{2}e^{tz_{i}}$,

one can check that

$Q_{2}(t,z_{1},\cdots,z_{n})=Q_{0}(t,z_{1},\cdots,z_{n})R_{2}(t,z_{1},\cdots,z_{n})-Q_{1}^{2}(t,z_{1},\cdots,z_{n})$.

In the following we will often fix $t=t_{*}>0$, so we will frequently write

$Q_{0}(t_{*},z_{1},z_{2},\cdots,z_{n})=Q_{0}(z_{1},z_{2},\cdots,z_{n})$

with $t_{*}$ omitted for convenience. Similar notations apply to other terms
above. Let

(5.4) $Q_{0}(z_{1},\cdots,z_{n})=\sum_{i=1}^{n}e^{t_{*}z_{i}}=e^{a_{0}},$
(5.5)
$Q_{1}(z_{1},\cdots,z_{n})=\sum_{i=1}^{n}z_{i}e^{t_{*}z_{i}}=a_{1}e^{a_{0}}$

be two equations with unknowns $\\{z_{1},z_{2},\cdots,z_{n}\\}$ for fixed
$t_{*}>0,(a_{0},a_{1})\in\mathbb{R}^{2}$ and some $n\geq 2$. Let

$\Gamma_{\ref{eq35}}^{n}=\\{(z_{1},z_{2},\cdots,z_{n})\in\mathbb{R}^{n}:z_{1},z_{2},\cdots,z_{n}\mbox{
satisfy }(\ref{eq35})\\}$

and

$\Gamma_{\ref{eq36}}^{n}=\\{(z_{1},z_{2},\cdots,z_{n})\in\mathbb{R}^{n}:z_{1},z_{2},\cdots,z_{n}\mbox{
satisfy }(\ref{eq36})\\}$.

They are both $n-1$ dimensional smooth hypersurfaces. We first present readers
with the following result on fitting an analytic function

$a_{0}+a_{1}(t-t_{*})+O((t-t_{*})^{2})$

with $t_{*},a_{0},a_{1}$ subject to (1.5) around some fixed $t_{*}>0$ by
pressures of locally constant potentials on general shift spaces of finite
type.

######  5.6 Theorem.

Let $t_{*}>0,(a_{0},a_{1})\in\mathbb{R}^{2},n\geq 2$ satisfying (1.5) and

(5.6) $\cfrac{a_{0}-\log n}{t_{*}}<a_{1}.$

Then there exists some $\delta_{n}>0$ and some sequence
$\\{r_{i,n}\\}_{i=1}^{n}\subset\mathbb{R}$, such that the locally constant
potential

$\phi(x)=r_{x_{0}}$

for $x=\cdots x_{-1}x_{0}x_{1}\cdots\in[x_{0}]$ on the full shift space
$\Lambda_{n}^{\mathbb{Z}}$ satisfies

$P(t\phi)=a_{0}+a_{1}(t-t_{*})+O((t-t_{*})^{2})$

on $[t_{*}-\delta_{n},t_{*}+\delta_{n}]$.

###### Proof.

In fact it suffices for us to show that the system of equations

$\left\\{\begin{array}[]{ll}(\ref{eq35}),\\\ (\ref{eq36})\end{array}\right.$

with unknowns $\\{z_{1},z_{2},\cdots,z_{n}\\}$ admits a solution under
conditions of the theorem. Without loss of generality we assume

(5.7) $z_{1}\leq z_{2}\leq\cdots\leq z_{n}.$

Under this assumption, it is easy to see that

$\cfrac{a_{0}-\log n}{t_{*}}\leq z_{n}<\cfrac{a_{0}}{t_{*}}$.

Now we estimate the values of $Q_{1}(z_{1},\cdots,z_{n})$ with $z_{n}$
approaching the terminals. When $z_{n}$ approaches the right terminal from
below, we have

$\lim_{(z_{1},z_{2},\cdots,z_{n})\in\Gamma_{\ref{eq35}}^{n},\
z_{n}\nearrow\frac{a_{0}}{t_{*}}}Q_{1}(z_{1},\cdots,z_{n})=\cfrac{a_{0}}{t_{*}}e^{a_{0}}>a_{1}e^{a_{0}}$

in virtue of (1.5). When $z_{n}$ approaches the left terminal from above, we
have

$\lim_{(z_{1},z_{2},\cdots,z_{n})\in\Gamma_{\ref{eq35}}^{n},\
z_{n}\searrow\frac{a_{0}}{t_{*}}}Q_{1}(z_{1},\cdots,z_{n})=\cfrac{a_{0}-\log
n}{t_{*}}e^{a_{0}}<a_{1}e^{a_{0}}$

in virtue of (5.6). Since $\Gamma_{\ref{eq35}}^{n}$ is a smooth hypersurface,
by the mean value theorem, there exists some
$(r_{1,n},r_{2,n},\cdots,r_{n,n})\in\Gamma_{\ref{eq35}}^{n}$ satisfying (5.4)
and (5.5) simultaneously. At last, for $x=\cdots
x_{-1}x_{0}x_{1}\cdots\in[x_{0}]$ on the full shift space
$\Lambda_{n}^{\mathbb{Z}}$, let

$\phi(x)=r_{x_{0},n}$

be the locally constant potential. As $P(t\phi)$ is analytic, there exists
some $\delta_{n}>0$ such that

$P(t\phi)=a_{0}+a_{1}(t-t_{*})+O((t-t_{*})^{2})$

for $t\in[t_{*}-\delta_{n},t_{*}+\delta_{n}]$. ∎

######  5.7 Remark.

The core step in the proof of Theorem 5.6 is in fact finding the extremes of
the function $Q_{1}(z_{1},\cdots,z_{n})$ subject to (5.4), (1.5) and (5.6).
One can detect the points of extremes by the Karush-Kuhn-Tucker (KKT)
conditions [Kar, KT], which generalizes the method of Lagrange multipliers by
allowing inequality subjections.

Be careful that those $\\{r_{i,n}\\}_{i=1}^{n}$ all depend on $n$ in fact.
Theorem 5.6 induces the following interesting flexibility result on fitting
certain analytic functions locally by pressures of locally constant potentials
on general shift space of finite type.

######  5.8 Corollary.

Let $t_{*}>0$ and $(a_{0},a_{1})\in\mathbb{R}^{2}$ satisfy (1.5). Then there
exists some $N\in\mathbb{N}$, such that for any $n\geq N$, there exist some
some $\delta_{n}>0$ and some sequence
$\\{r_{i,n}\\}_{i=1}^{n}\subset\mathbb{R}$, such that the locally constant
potential

$\phi(x)=r_{x_{0},n}$

for $x=\cdots x_{-1}x_{0}x_{1}\cdots\in[x_{0}]$ on the full shift space
$\Lambda_{n}^{\mathbb{Z}}$ satisfies

$P(t\phi)=a_{0}+a_{1}(t-t_{*})+O((t-t_{*})^{2})$

on $[t_{*}-\delta_{n},t_{*}+\delta_{n}]$.

###### Proof.

Under conditions of the corollary, for the given values $t_{*},a_{0},a_{1}$
satisfying (1.5), choose $N\in\mathbb{N}$ large enough such that

$\cfrac{a_{0}-\log N}{t_{*}}<a_{1}$.

This means that for any $n>N$ condition (5.6) is satisfied for
$t_{*},a_{0},a_{1},n$. Then the conclusion follows from Theorem 5.6. ∎

Note that on some particular symbolic spaces Theorem 5.6 and 5.8 may be
trivial. For example, for given $(t_{*},a_{0},a_{1})\in\mathbb{R}^{3}$ without
any subjections, by choosing $\beta=e^{a_{0}-t_{*}a_{1}}$, consider the
constant potential

$\phi(x)=a_{1}$

on the $\beta$-shift space with symbols
$\\{0,1,\cdots,\lfloor\beta\rfloor\\}$. It is easy to see that

$P(t\phi)=a_{0}-t_{*}a_{1}+a_{1}t=a_{0}+a_{1}(t-t_{*})$

on $(-\infty,\infty)$. However, our results guarantee conclusions on general
shift spaces.

From now on we go towards the proof of Theorem 1.8. For fixed
$t_{*}>0,(a_{0},a_{1})\in\mathbb{R}^{2}$ and $n\geq 3$, let

$\Gamma_{\ref{eq35},\ref{eq36}}^{n}=\Gamma_{\ref{eq35}}^{n}\cap\Gamma_{\ref{eq36}}^{n}=\\{(z_{1},z_{2},\cdots,z_{n})\in\mathbb{R}^{n}:z_{1},z_{2},\cdots,z_{n}\mbox{
satisfy }(\ref{eq35})\mbox{ and }(\ref{eq36})\\}$.

We describe some topological properties of the set
$\Gamma_{\ref{eq35},\ref{eq36}}^{n}$ in the following result.

######  5.9 Lemma.

For fixed $t_{*}>0,(a_{0},a_{1})\in\mathbb{R}^{2}$ subject to (1.5) and $n\geq
3$, in case $\Gamma_{\ref{eq35},\ref{eq36}}^{n}\neq\emptyset$ and
$a_{1}\neq\cfrac{a_{0}-\log n}{t_{*}}$, it is a compact $(n-2)$-dimension
smooth manifold.

###### Proof.

The Jacobian of the functions $Q_{0}(z_{1},\cdots,z_{n})-e^{a_{0}}$ and
$Q_{1}(z_{1},\cdots,z_{n})-a_{1}e^{a_{0}}$ with respect to
$z_{1},z_{2},\cdots,z_{n}$ is

$J=\begin{pmatrix}t_{*}e^{t_{*}z_{1}}&t_{*}e^{t_{*}z_{2}}&\cdots&t_{*}e^{t_{*}z_{n}}\\\
e^{t_{*}z_{1}}+t_{*}z_{1}e^{t_{*}z_{1}}&e^{t_{*}z_{2}}+t_{*}z_{2}e^{t_{*}z_{2}}&\cdots&e^{t_{*}z_{n}}+t_{*}z_{n}e^{t_{*}z_{n}}\end{pmatrix}.$

Its rank is strictly less than $2$ if and only if

$z_{1}=z_{2}=\cdots=z_{n}$.

Since $a_{1}\neq\cfrac{a_{0}-\log n}{t_{*}}$, this is excluded from points in
$\Gamma_{\ref{eq35},\ref{eq36}}$. By the implicit function theorem [Lan], if
$\Gamma_{\ref{eq35},\ref{eq36}}^{n}$ is not empty, it is an $(n-2)$-dimension
smooth manifold locally. The gradient of the function
$Q_{0}(z_{1},\cdots,z_{n})-e^{a_{0}}$ is

$\bigtriangledown(Q_{0}(z_{1},\cdots,z_{n})-e^{a_{0}})=(t_{*}e^{t_{*}z_{1}},t_{*}e^{t_{*}z_{2}},\cdots,t_{*}e^{t_{*}z_{n}})$,

whose individual components will always be strictly positive. The gradient of
the function $Q_{1}(z_{1},\cdots,z_{n})-a_{1}e^{a_{0}}$ is

$\bigtriangledown(Q_{1}(z_{1},\cdots,z_{n})-a_{1}e^{a_{0}})=(e^{t_{*}z_{1}}+t_{*}z_{1}e^{t_{*}z_{1}},e^{t_{*}z_{2}}+t_{*}z_{2}e^{t_{*}z_{2}},\cdots,e^{t_{*}z_{n}}+t_{*}z_{n}e^{t_{*}z_{n}})$,

with the $i$-th individual component vanishes if and only if
$z_{i}=-\cfrac{1}{t_{*}}$ for $1\leq i\leq n$. So $\Gamma_{\ref{eq35}}^{n}$
and $\Gamma_{\ref{eq36}}^{n}$ cannot be tangent to each other. Moreover, note
that

$e^{t_{*}z_{i}}+t_{*}z_{i}e^{t_{*}z_{i}}>0$

if $z_{i}>-\cfrac{1}{t_{*}}$ while

$e^{t_{*}z_{i}}+t_{*}z_{i}e^{t_{*}z_{i}}<0$

if $z_{i}<-\cfrac{1}{t_{*}}$ for any $1\leq i\leq n$. These force the
intersection of zeros of the two functions
$Q_{0}(z_{1},\cdots,z_{n})-e^{a_{0}}$ and
$Q_{1}(z_{1},\cdots,z_{n})-a_{1}e^{a_{0}}$ to be connected, if the
intersection is not empty. This implies $\Gamma_{\ref{eq35},\ref{eq36}}$ is a
manifold globally in case of being nonempty.
$\Gamma_{\ref{eq35},\ref{eq36}}^{n}$ is compact since it is a bounded set. ∎

Let

$\Gamma_{\ref{eq35},1,2,1}^{3}=\\{(z_{1},z_{2},z_{3})\in\mathbb{R}^{3}:z_{1},z_{2},z_{3}\mbox{
satisfy }e^{z_{1}}+e^{z_{2}}+e^{z_{3}}=e^{2}\\}$

and

$\Gamma_{\ref{eq36},1,2,1}^{3}=\\{(z_{1},z_{2},z_{3})\in\mathbb{R}^{3}:z_{1},z_{2},z_{3}\mbox{
satisfy }z_{1}e^{z_{1}}+z_{2}e^{z_{2}}+z_{3}e^{z_{3}}=e^{2}\\}$

be the corresponding surfaces with $t_{*}=1,a_{0}=2,a_{1}=1$. Figure 2 depicts
parts of the two $2$-dimension surfaces, whose intersection will be a
$1$-dimension smooth curve.

Figure 2. $\Gamma_{\ref{eq35},1,2,1}^{3}$ (green) and
$\Gamma_{\ref{eq36},1,2,1}^{3}$ (red)

Equipped with all the above results, now we are ready to prove Theorem 1.8.

###### Proof of Theorem 1.8.

First, for the given $t_{*}>0$ and $(a_{0},a_{1})\in\mathbb{R}^{2}$ satisfying
(1.5), if $n$ is large enough, $\Gamma_{\ref{eq35},\ref{eq36}}^{n}$ is not
empty according to Corollary 5.8. So $\Gamma_{\ref{eq35},\ref{eq36}}^{n}$ is a
compact $(n-2)$-dimension smooth manifold for $n$ large enough. In the
following we always assume $n$ is large enough. Now let

$m_{t_{*},a_{0},a_{1},n}=\min\Big{\\{}\cfrac{R_{2}(t_{*},z_{1},z_{2},\cdots,z_{n})}{e^{a_{0}}}-a_{1}^{2}:(z_{1},z_{2},\cdots,z_{n})\in\Gamma_{\ref{eq35},\ref{eq36}}^{n}\Big{\\}}$

while

(5.8)
$M_{t_{*},a_{0},a_{1},n}=\max\Big{\\{}\cfrac{R_{2}(t_{*},z_{1},z_{2},\cdots,z_{n})}{e^{a_{0}}}-a_{1}^{2}:(z_{1},z_{2},\cdots,z_{n})\in\Gamma_{\ref{eq35},\ref{eq36}}^{n}\Big{\\}}.$

For any $m_{t_{*},a_{0},a_{1},n}\leq a_{2}\leq M_{t_{*},a_{0},a_{1},n}$, since
$\Gamma_{\ref{eq35},\ref{eq36}}^{n}$ is a smooth manifold, there exist
$\\{c_{i,n}\\}_{i=1}^{n}\subset\mathbb{R}$, such that
$(c_{1,n},c_{2,n},\cdots,c_{n,n})$ satisfies $(\ref{eq35}),(\ref{eq36})$ and

(5.9)
$a_{2}=\cfrac{Q_{2}(t_{*},c_{1,n},\cdots,c_{n,n})}{Q_{0}^{2}(t_{*},c_{1,n},\cdots,c_{n,n})}=\cfrac{R_{2}(t_{*},c_{1,n},\cdots,c_{n,n})}{e^{a_{0}}}-a_{1}^{2}$

simultaneously. Now let

$\phi(x)=c_{x_{0},n}$

for $x=\cdots x_{-1}x_{0}x_{1}\cdots\in[x_{0}]$ on the full shift space
$\Lambda_{n}^{\mathbb{Z}}$. It is a locally constant potential. According to
(5.3) and (5.9), we have

(5.10)
$P^{\prime\prime}(t_{*}\phi)=\cfrac{Q_{2}(t_{*},c_{1,n},\cdots,c_{n,n})}{Q_{0}^{2}(t_{*},c_{1,n},\cdots,c_{n,n})}=a_{2}.$

Since $(c_{1,n},c_{2,n},\cdots,c_{n,n})$ satisfies $(\ref{eq35})$ and
$(\ref{eq36})$, we have

(5.11)
$P(t_{*}\phi)=\cfrac{Q_{2}(t_{*},c_{1,n},\cdots,c_{n,n})}{Q_{0}^{2}(t_{*},c_{1,n},\cdots,c_{n,n})}=a_{0}$

while

(5.12)
$P^{\prime}(t_{*}\phi)=\cfrac{Q_{2}(t_{*},c_{1,n},\cdots,c_{n,n})}{Q_{0}^{2}(t_{*},c_{1,n},\cdots,c_{n,n})}=a_{1}.$

Note that $P(t\phi)$ is analytic with respect to $t$ on $(\alpha,\infty)$ for
any $\alpha>0$, so there exists some $\delta_{n}>0$, such that (1.6) holds on
$[t_{*}-\delta_{n},t_{*}+\delta_{n}]$, considering (5.10), (5.11) and (5.12).
∎

In the following we illustrate some dependent relationship between

$\\{m_{t_{*},a_{0},a_{1},n},M_{t_{*},a_{0},a_{1},n}\\}_{n\in\mathbb{N}}$

and some particular $t_{*},a_{0},a_{1},n$ satisfying (1.5). There should be
some universal relationship between them, while we hope the following
observations will provide some hints. The first one is that it is possible for
$m_{t_{*},a_{0},a_{1},n}=0$ for some $t_{*},a_{0},a_{1},n$.

######  5.10 Proposition.

Let $t_{*}>0$ and $(a_{0},a_{1})\in\mathbb{R}^{2}$ satisfy (1.5). Then
$m_{t_{*},a_{0},a_{1},n}=0$ for $n\geq 2$ if and only if

(5.13) $a_{1}=\cfrac{a_{0}-\log n}{t_{*}}.$

###### Proof.

Note that $m_{t_{*},a_{0},a_{1},n}=0$ is equivalent to say that there exists
some locally constant potential $\phi$ on $\Lambda_{n}^{\mathbb{Z}}$ such that
$P^{\prime\prime}(t_{*}\phi)=0$ according to Theorem 1.8. By [PP, Proposition
4.12], this happens if and only if $\phi$ is a constant potential on
$\Lambda_{n}^{\mathbb{Z}}$. In this case we have

$\phi(x)=\cfrac{a_{0}-\log n}{t_{*}}$

for any $x\in\Lambda_{n}^{\mathbb{Z}}$, which implies (5.13). ∎

This result does not tell things about the sequence

$\\{m_{t_{*},a_{0},a_{1},n}\\}_{\ n\in\mathbb{N}\mbox{ large enough }}$

for given $t_{*},a_{0},a_{1}$, since (5.13) will never be true for any $n$
large enough for fixed $t_{*},a_{0},a_{1}$. The following result describes
some limit behaviour of the sequence

$\\{M_{t_{*},a_{0},a_{1},n}\\}_{\ n\in\mathbb{N}\mbox{ large enough }}$

for $t_{*}=1,a_{0}=2,a_{1}=1$.

######  5.11 Proposition.

Let $t_{*}=1,a_{0}=2,a_{1}=1$, in symbols of Theorem 1.8, we have

(5.14) $\lim_{n\rightarrow\infty}M_{1,2,1,n}=\infty.$

To justify Proposition 5.11, we first illustrate some basic properties about
the function $ze^{t_{*}z}$ for $t_{*}>0$.

######  5.12 Lemma.

For $t_{*}>0$, $ze^{t_{*}z}$ is strictly decreasing on
$(-\infty,-\cfrac{1}{t_{*}})$, strictly increasing on
$(-\cfrac{1}{t_{*}},\infty)$, while it attains its minimum
$-\cfrac{1}{t_{*}}e^{-1}$ at $z=-\cfrac{1}{t_{*}}$. It admits one and only one
inflection in $(-\infty,-\cfrac{1}{t_{*}})$.

###### Proof.

One can check these conclusions by some direct computations on the first and
second derivatives of the function $ze^{t_{*}z}$. ∎

In Figure 3 we depict the graph of $\varsigma(z)=ze^{z}$.

Figure 3. Graph of $\varsigma(z)=ze^{z}$ Figure 4. $\Gamma_{\ref{eq46}}$ and
$\Gamma_{\ref{eq47}}$

###### Proof of Proposition 5.11.

Since we are considering the limit behaviour of $M_{1,2,1,n}$, we always
assume $n$ is large enough throughout the proof. Now consider the following
two equations

(5.15) $(n-1)e^{z_{a}}+e^{z_{b}}=e^{2}$

and

(5.16) $(n-1)z_{a}e^{z_{a}}+z_{b}e^{z_{b}}=e^{2}$

with unknowns $z_{a},z_{b}$. Let

$\Gamma_{\ref{eq46}}=\\{(z_{a},z_{b})\in\mathbb{R}^{2}:z_{a},z_{b}\mbox{
satisfy }(\ref{eq46})\\}$

and

$\Gamma_{\ref{eq47}}=\\{(z_{a},z_{b})\in\mathbb{R}^{2}:z_{a},z_{b}\mbox{
satisfy }(\ref{eq47})\\}$.

We describe the graph of $\Gamma_{\ref{eq46}}$ and $\Gamma_{\ref{eq47}}$
separately in the following. $\Gamma_{\ref{eq46}}$ is a $1$ dimensional smooth
curve with two asymptotes $z_{a}=2-\log(n-1)$ and $z_{b}=2$. It is strictly
decreasing when we consider the curve as the graph of the function

$z_{b}=\log\big{(}e^{2}-(n-1)e^{z_{a}}\big{)}$

for $z_{a}\in(-\infty,2-\log(n-1))$. $\Gamma_{\ref{eq47}}$ is also a $1$
dimensional smooth curve with two asymptotes
$z_{a}=\varsigma^{-1}(\cfrac{e^{2}}{n-1})$ and $z_{b}=\varsigma^{-1}(e^{2})$.
When we consider the $\Gamma_{\ref{eq47}}$ as the graph of the function

$z_{b}=\eta(z_{a})$

as the implicit function induced by $(\ref{eq47})$, it is strictly increasing
for $z_{a}\in(-\infty,-1)$, strictly decreasing for
$z_{a}\in(-1,\varsigma^{-1}(\cfrac{e^{2}}{n-1}))$, with its maximum
$\varsigma^{-1}(e^{2}+(n-1)e^{-1})$ attained at $z_{a}=-1$. Let
$\varsigma_{l}^{-1}(-\cfrac{e^{2}}{n-1})$ be the smaller one of the two
intersections of $z_{b}=2$ and $\Gamma_{\ref{eq47}}$, then
$\Gamma_{\ref{eq46}}$ and $\Gamma_{\ref{eq47}}$ must intersection at some
unique point $c_{a,n}\in(-\infty,\varsigma_{l}^{-1}(-\cfrac{e^{2}}{n-1}))$.
Obviously

$\lim_{n\rightarrow\infty}c_{a,n}=-\infty$

since
$\lim_{n\rightarrow\infty}\varsigma_{l}^{-1}(-\cfrac{e^{2}}{n-1})=-\infty$.
Now we analyse the order of $c_{a,n}$ with respect to $n$ as
$n\rightarrow\infty$. Let

$z_{a,n}=-\log n-\log\log n+\log 1-1$.

One can check that

$\lim_{n\rightarrow\infty,z_{b}\rightarrow
2}((n-1)e^{z_{a,n}}+e^{z_{b}})=e^{2}$

while

$\lim_{n\rightarrow\infty,z_{b}\rightarrow
2}((n-1)z_{a,n}e^{z_{a,n}}+z_{b}e^{z_{b}})=e^{2}$.

These imply that

$c_{a,n}=-\log n-\log\log n+o(\log\log n)$.

Note that
$(c_{a,n},c_{a,n},\cdots,c_{a,n},\eta(c_{a,n}))\in\Gamma_{\ref{eq35},\ref{eq36}}^{n}$
for $t_{*}=1,a_{0}=2,a_{1}=1$. Now

$\begin{array}[]{ll}&R_{2}(c_{a,n},c_{a,n},\cdots,c_{a,n},\eta(c_{a,n}))\vspace{3mm}\\\
=&(n-1)c_{a,n}^{2}e^{c_{a,n}}+(\eta(c_{a,n}))^{2}e^{\eta(c_{a,n})}\vspace{3mm}\\\
=&(n-1)(-\log n-\log\log n+o(\log\log n))^{2}e^{-\log n-\log\log n+o(\log\log
n)}+4e^{2}+o(1)\vspace{3mm}\\\ =&\log n+o(\log n),\end{array}$

from which it is easy to see that

$\lim_{n\rightarrow\infty}R_{2}(c_{a,n},c_{a,n},\cdots,c_{a,n},\eta(c_{a,n}))=\infty$.

This forces

$\lim_{n\rightarrow\infty}M_{1,2,1,n}=\infty$,

considering (5.8).

∎

We provide the readers with the curves $\Gamma_{\ref{eq46}}$ and
$\Gamma_{\ref{eq47}}$ in Figure 4. Obviously some more general conclusions are
available if one considers variations of the parameters $t_{*},a_{0},a_{1}$ in
Proposition 5.11. At last we provide the readers with some solutions
$\\{c_{a,n}\\}_{n\in\mathbb{N}}$ and $\\{\eta(c_{a,n})\\}_{n\in\mathbb{N}}$ in
Table 3, from which one can see the order of decay and increase of the
sequences with respect to $n$ clearly.

Table 3. $\\{c_{a,n}\\}_{n\in\mathbb{N}}$ and $\\{\eta(c_{a,n})\\}_{n\in\mathbb{N}}$ $n$ | $c_{a,n}$ | $\eta(c_{a,n})$
---|---|---
$10$ | -1.8599539391797653780996686364493 | 1.7634042477581860636342812520981
$10^{2}$ | -4.6278529940301947157458180305676 | 1.8580906928560505140960875180438
$10^{3}$ | -7.2278923365046354303919671475052 | 1.8965708210067454817129699066334
$10^{4}$ | -9.7529279223041958189401940128674 | 1.9180710389285259082138396366755
$10^{5}$ | -12.23426184122178540565187685582 | 1.9319494203818796717151866525306
$10^{6}$ | -14.686689485112383196253350885528 | 1.941701042038176132682488585943
$10^{7}$ | -17.118475509130338419321449219176 | 1.9489507180131363431129601417792
$10^{8}$ | -19.534737736752111249670741176574 | 1.9545628133690736391913141129777
$10^{9}$ | -21.938877884281897893422087428599 | 1.9590417833080193886068703580662
$10^{10}$ | -24.333277592346602338263750350022 | 1.9627027620469153955488959845337
$10^{11}$ | -26.719672172461371813735932628894 | 1.9657531814729595378854181456218
$10^{12}$ | -29.099366670257435261982274861811 | 1.9683353707111573738492465130807
$10^{13}$ | -31.473368167571030624456199153849 | 1.970550350496947761285545176838
$10^{14}$ | -33.842470627269595326611535858951 | 1.9724718685216929582206115029034
$10^{15}$ | -36.20731141238751139407393422892 | 1.9741550583546827046855344007126
$10^{16}$ | -38.568410155198951836337896822881 | 1.97564198636943790477268372057
$10^{17}$ | -40.926196222869058989174011616314 | 1.9769653208730088904749619599928
$10^{18}$ | -43.28102858421294787781225809291 | 1.9781508271703613365389080750692
$10^{19}$ | -45.633210475623427729647938869856 | 1.9792191056459012534062976747755
$10^{20}$ | -47.983000423353389741328990557576 | 1.9801868284846851379610473178804
$10^{21}$ | -50.330620660008332271820694306839 | 1.9810676363715292020369862557429
$10^{22}$ | -52.676263643082855194671803053742 | 1.9818727996772032079642260800619
$10^{23}$ | -55.020097168291592849066888176454 | 1.9826117134133018944596936081392
$10^{24}$ | -57.362268427077060922578379063246 | 1.9832922728467949817312209115653
$10^{25}$ | -59.702907260160132201351723856461 | 1.9839211621102961084222105523408
$10^{26}$ | -62.042128791447074538616865826092 | 1.9845040784885601043186175801529
$10^{27}$ | -64.380035579030470553978577616248 | 1.9850459085371711281404342988732
$10^{28}$ | -66.716719386002755126963619613768 | 1.9855508677057357884921072471682
$10^{29}$ | -69.052262649137714881922574449762 | 1.9860226120088820356292880321385
$10^{30}$ | -71.386739705385277326962820249044 | 1.9864643280735340181774784668139
$10^{31}$ | -73.7202178226698188293210417966 | 1.9868788063025456510398996161088
$10^{32}$ | -76.052758071376257724956806229201 | 1.9872685007417625212636588061233
$10^{33}$ | -78.384416065240707497606345034329 | 1.9876355783911370649894789906831
$10^{34}$ | -80.715242594490126828808238297291 | 1.9879819600725889180558042388717
$10^{35}$ | -83.045284169538297201269228695051 | 1.9883093544968926933418986392956
$10^{36}$ | -85.374583490011093204926910773042 | 1.9886192868162322855194994642595
$10^{37}$ | -87.703179851099500408441885242821 | 1.9889131226778662869710690898493
$10^{38}$ | -90.031109497045012553979249690171 | 1.9891920885858755212588911444123
$10^{39}$ | -92.358405929815521230622254914238 | 1.9894572892164831961499157326311
$10^{40}$ | -94.685100179630439886817678169988 | 1.9897097222064583485549741614556

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