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# Nearly toroidal, periodic and quasi-periodic motions
of fluid particles driven by the Gavrilov solutions
of the Euler equations

Pietro Baldi

Abstract. We consider the smooth, compactly supported solutions of the steady
3D Euler equations of incompressible fluids contructed by Gavrilov in 2019,
and we study the corresponding fluid particle dynamics. This is an ode
analysis, which contributes to the description of Gavrilov’s vector field.

###### Contents

  1. 1 Introduction and main result
    1. 1.1 The Gavrilov solutions of the steady Euler equations
    2. 1.2 Main result: description of the fluid particle dynamics
    3. 1.3 Related literature
  2. 2 Dual role of the parameter $R$ in Gavrilov’s solutions
  3. 3 Conjugation to a linear flow on ${\mathbb{T}}^{2}$
    1. 3.1 Cilindrical coordinates
    2. 3.2 Elimination of the factor $1/\rho$ and canonical Hamiltonian structure
    3. 3.3 Symplectic polar coordinates in the radial-vertical plane
    4. 3.4 Level sets
    5. 3.5 Integration of the system in the radial-vertical plane
    6. 3.6 Rotation period in the radial-vertical plane
    7. 3.7 Angle-action variables in the radial-vertical plane
    8. 3.8 Reduction to a constant rotation in the tangential direction
    9. 3.9 Ratio of the two rotation periods
    10. 3.10 Motion of the fluid particles
    11. 3.11 The pressure in terms of the action
  4. 4 Taylor expansions and transversality
    1. 4.1 Expansion of $\alpha$
    2. 4.2 Expansion of $\alpha_{2}$
    3. 4.3 Expansion of $\gamma_{c}(\vartheta)$
    4. 4.4 Expansion of the average of $\gamma_{c}(\vartheta)$ and $\partial_{c}\gamma_{c}(\vartheta)$
    5. 4.5 Expansion of $J(c)$
    6. 4.6 Expansion of $h(I)$ and $\Omega_{1}(I)$
    7. 4.7 Expansion of the frequency ratio $\Omega_{2}(I)/\Omega_{1}(I)$
    8. 4.8 Expansion of $\Phi(\sigma,\beta,I)$
    9. 4.9 Smallness conditions

##  1 Introduction and main result

In the remarkable paper [7], Gavrilov proved the existence of a nontrivial
solution of $C^{\infty}$ class, with compact support, of the steady Euler
equations of incompressible fluids in ${\mathbb{R}}^{3}$. The result in [7] is
important and surprising, because previously, on the basis of some negative
partial results, it was conjectured that compactly supported, nontrivial,
smooth solutions of the 3D steady Euler equations cannot exist: see the clear
explanation at the beginning of [4] and the general discussion about existence
of compactly supported smooth solutions in [10]. In addition, another reason
of interest for the fruitful construction of [7] is that recently it has been
used as a building block to produce other interesting solutions, both
stationary and time-dependent, of the Euler equations of fluid dynamics, see
Section 1.3 below.

Now suppose that a fluid moves according to the Gavrilov solution of the Euler
equations, that is, suppose that the fluid particles are driven by Gavrilov’s
velocity vector field. Which movement of the fluid do we observe? Of course
the particles outside the support of the vector field do not move at all, but
how do they move in the region where the field is nonzero?

In the present paper we deal with this question. It turns out that every fluid
particle travels along a trajectory that lies on a nearly toroidal surface,
which is a level set of the pressure. The motion of every fluid particle is
periodic or quasi-periodic in time; we prove that there are both periodic and
quasi-periodic motions, and the value of the pressure determines whether the
trajectories on its level set are all periodic or all quasi-periodic.

In fact, the system of differential equations in ${\mathbb{R}}^{3}$ describing
the motion of the fluid particles turns out to be integrable, as it can be
transformed into the system of a Hamiltonian system of one degree of freedom
and a third equation that can be directly solved by integration. We write the
Hamiltonian system in angle-action coordinates $(\sigma,I)$, and prove that
there exists a change of variables on a neighborhood of the support of
Gavrilov’s vector field such that the equations of motion in the new
coordinates $(\sigma,\beta,I)$ becomes

$\dot{\sigma}=\Omega_{1}(I),\quad\ \dot{\beta}=\Omega_{2}(I),\quad\
\dot{I}=0,$

where $\sigma$ and $\beta$ are angle variables rotating with constant angular
velocities $\Omega_{1}(I)$, $\Omega_{2}(I)$, and $I$ is a constant action
variable, which is, in fact, a reparametrization of the pressure. The full
statement is in Theorem 1.1.

###  1.1 The Gavrilov solutions of the steady Euler equations

In the main part of the construction in [7], given any $R>0$, the circle

$\mathcal{C}:=\\{(x,y,z)\in{\mathbb{R}}^{3}:x^{2}+y^{2}=R^{2},\ z=0\\}$

in ${\mathbb{R}}^{3}$ is considered, and, in an open neighborhood
$\mathcal{N}$ of $\mathcal{C}$, two functions $U,P$ are defined,
$U:\mathcal{N}\to{\mathbb{R}}^{3}$ and $P:\mathcal{N}\to{\mathbb{R}}$, both in
$C^{\infty}(\mathcal{N}\setminus\mathcal{C})$, solving the steady Euler
equations

$U\cdot\nabla U+\nabla P=0,\quad\ \mathrm{div}\,U=0$ (1.1)

in $\mathcal{N}\setminus\mathcal{C}$, together with the fundamental
“localizability condition”

$U\cdot\nabla P=0.$ (1.2)

As a final step of the proof, the functions $U,P$ are multiplied by smooth
cut-off functions to obtain $C^{\infty}({\mathbb{R}}^{3})$ functions
$\tilde{U},\tilde{P}$, where $\tilde{U}$ and $\nabla\tilde{P}$ have compact
support contained in $\mathcal{N}\setminus\mathcal{C}$, solving (1.1) (and
also (1.2)) in ${\mathbb{R}}^{3}$.

Let us be more precise. Denote $\rho:=\sqrt{x^{2}+y^{2}}$. For
$\delta\in(0,R)$, let

$\mathcal{N}=\\{(x,y,z)\in{\mathbb{R}}^{3}:(\rho-R)^{2}+z^{2}<\delta^{2}\\}.$
(1.3)

In $\mathcal{N}$, the solution $(U,P)$ of [7] is given by

$U(x,y,z)=u_{\rho}(\rho,z)e_{\rho}(x,y)+u_{\varphi}(\rho,z)e_{\varphi}(x,y)+u_{z}(\rho,z)e_{z},\quad
P(x,y,z)=p(\rho,z),$ (1.4)

where

$\displaystyle e_{\rho}(x,y)$ $\displaystyle=\frac{1}{\rho}(x,y,0),$
$\displaystyle\quad\ e_{\varphi}(x,y)$ $\displaystyle=\frac{1}{\rho}(-y,x,0),$
$\displaystyle\quad\ e_{z}$ $\displaystyle=(0,0,1),$ $\displaystyle
u_{\rho}(\rho,z)$ $\displaystyle=\frac{\partial_{z}p(\rho,z)}{\rho},$
$\displaystyle\quad\ u_{\varphi}(\rho,z)$
$\displaystyle=\frac{b(\rho,z)}{\rho},$ $\displaystyle\quad\ u_{z}(\rho,z)$
$\displaystyle=-\frac{\partial_{\rho}p(\rho,z)}{\rho},$ $\displaystyle
b(\rho,z)$ $\displaystyle=\frac{R^{3}}{4}\sqrt{H(a(\rho,z))},$
$\displaystyle\quad\ p(\rho,z)$ $\displaystyle=\frac{R^{4}}{4}a(\rho,z),$
$\displaystyle\quad\ a(\rho,z)$
$\displaystyle=\alpha\Big{(}\frac{\rho}{R},\frac{z}{R}\Big{)},$ (1.5)

and $\alpha,H$ are functions defined in [7] in terms of solutions of certain
differential equations; $H(0)=0$, and $H$ is analytic in a neighborhood of
$0$; $\alpha$ has a strict local minimum at $(1,0)$, with $\alpha(1,0)=0$, and
it is analytic in a neighborhood of $(1,0)$. Hence $\alpha$ and $H\circ\alpha$
are both well-defined and analytic in a disc of ${\mathbb{R}}^{2}$ of center
$(1,0)$ and radius $r_{0}$, for some universal constant $r_{0}>0$ (where
“universal” means that $r_{0}$ does not depend on anything). If $\delta$ in
(1.3) satisfies

$\delta\leq r_{0}R,$

then $a(\rho,z)$ and $H(a(\rho,z))$, where $\rho=\sqrt{x^{2}+y^{2}}$, are
well-defined and analytic in $\mathcal{N}$ (note that, in $\mathcal{N}$, one
has $0<R-\delta<\rho<R+\delta$; in particular, $\rho$ is bounded away from
zero). Also, $b(\rho,z)$ is well-defined and continuous in $\mathcal{N}$, and
(because of the square root $\sqrt{H}$) it is analytic in
$\mathcal{N}\setminus\mathcal{C}$. Hence $P$ is analytic in $\mathcal{N}$,
while $U$ is continuous in $\mathcal{N}$ and analytic in
$\mathcal{N}\setminus\mathcal{C}$.

The solution $(\tilde{U},\tilde{P})$ in [7] is defined in $\mathcal{N}$ as

$\tilde{U}(x,y,z)=\omega(P(x,y,z))U(x,y,z),\quad\
\tilde{P}(x,y,z)=W(P(x,y,z)),$ (1.6)

where $\omega:{\mathbb{R}}\to{\mathbb{R}}$ is any $C^{\infty}$ function
vanishing outside the interval $[\varepsilon,2\varepsilon]$, with
$\varepsilon>0$ small enough, and $W:{\mathbb{R}}\to{\mathbb{R}}$ is a
primitive of $\omega^{2}$; for example,

$W(s)=\int_{0}^{s}\omega^{2}(\sigma)\,d\sigma.$ (1.7)

Then $(\tilde{U},\tilde{P})$ is extended to ${\mathbb{R}}^{3}$ by defining
$(\tilde{U},\tilde{P})=(0,W(2\varepsilon))$ in
${\mathbb{R}}^{3}\setminus\mathcal{N}$.

Note that $\tilde{U}$ and $\nabla\tilde{P}$ can be nonzero only in the set

$\mathcal{S}:=\\{(x,y,z)\in\mathcal{N}:\varepsilon<P(x,y,z)<2\varepsilon\\},$
(1.8)

and, if $\omega(s)$ is nonzero at some $s\in(\varepsilon,2\varepsilon)$, then
both $\tilde{U}$ and $\nabla\tilde{P}$ are actually nonzero at the
corresponding points in $\mathcal{S}$. Moreover, $P=0$ on the circle
$\mathcal{C}$, and, for $\varepsilon$ small enough, $P>3\varepsilon$ at all
points of $\mathcal{N}$ sufficiently far from $\mathcal{C}$; more precisely,
to fix the details of the construction, we introduce a parameter $\tau>0$ and
assume that

$P(x,y,z)>\tau\quad\
\forall(x,y,z)\in\mathcal{N}\setminus\mathcal{N}^{\prime}$ (1.9)

and $\tau\geq 3\varepsilon$, where

$\mathcal{N}^{\prime}:=\\{(x,y,z)\in{\mathbb{R}}^{3}:(\rho-R)^{2}+z^{2}<(\delta/4)^{2}\\}.$

Thus, the closure of $\mathcal{S}$ is contained in the open set

$\mathcal{S}^{*}:=\\{(x,y,z)\in\mathcal{N}:0<P(x,y,z)<\tau\\},$ (1.10)

and
$\mathcal{S}^{*}\subseteq(\mathcal{N}^{\prime}\setminus\mathcal{C})\subseteq(\mathcal{N}\setminus\mathcal{C})$.

###  1.2 Main result: description of the fluid particle dynamics

A preliminary, basic observation regarding the solutions of Section 1.1 is
that any such solution with a given $R>0$ can be obtained, by rescaling, from
another one having $R=1$. Even more, we show that Gavrilov’s solutions are a
1-parameter subset of a 2-parameter family of solutions, where $R$ plays a
dual role related to two different scaling invariances of the Euler equations,
both preserving the localizability condition (1.2). These basic observations
are in Section 2 (see Lemma 2.1). Thanks to these properties, we study the
motion of the fluid particles in the normalized case $R=1$; the motion for any
other $R>0$ is immediately obtained by rescaling the amplitude and the time
variable, as explained in Lemma 2.2.

To study the motion of the fluid particles driven by the Gavrilov vector field
$\tilde{U}$ means to study the solutions ${\mathbb{R}}\to{\mathbb{R}}^{3}$,
$t\mapsto(x(t),y(t),z(t))$ of the system

$(\dot{x}(t),\dot{y}(t),\dot{z}(t))=\tilde{U}(x(t),y(t),z(t)),$ (1.11)

which is an autonomous ode in ${\mathbb{R}}^{3}$. The dot above a function
denotes its time derivative. Before dealing with system (1.11), we recall some
definitions about quasi-periodic functions.

A vector $\Omega=(\Omega_{1},\ldots,\Omega_{n})\in{\mathbb{R}}^{n}$, $n\geq
1$, is said to be _rationally independent_ if $\Omega\cdot
k=\Omega_{1}k_{1}+\ldots+\Omega_{n}k_{n}$ is nonzero for all integer vectors
$k=(k_{1},\ldots,k_{n})\in\mathbb{Z}^{n}\setminus\\{0\\}$.

Given a set $X$, a function $v:{\mathbb{R}}\to X$, $t\mapsto v(t)$ is said to
be _quasi-periodic with frequency vector_
$\Omega=(\Omega_{1},\ldots,\Omega_{n})\in{\mathbb{R}}^{n}$, $n\geq 2$, if
$\Omega$ is rationally independent and there exists a function
$w:{\mathbb{R}}^{n}\to X$, $2\pi$-periodic in each real variable, such that
$v(t)=w(\Omega_{1}t,\ldots,\Omega_{n}t)$ for all $t\in{\mathbb{R}}$. Moreover,
to ensure that the number $n$ is not higher than necessary, we add the
condition that there does not exist any vector
$\tilde{\Omega}=(\tilde{\Omega}_{1},\ldots,\tilde{\Omega}_{m})\in{\mathbb{R}}^{m}$,
with $m<n$, and any function $\tilde{w}:{\mathbb{R}}^{m}\to X$,
$2\pi$-periodic in each real variable, such that
$v(t)=\tilde{w}(\tilde{\Omega}_{1}t,\ldots,\tilde{\Omega}_{m}t)$ for all
$t\in{\mathbb{R}}$.

For example, for $X={\mathbb{R}}$, $n=3$, and
$\Omega=(\Omega_{1},\Omega_{2},\Omega_{3})\in{\mathbb{R}}^{3}$ a rationally
independent vector, if
$w(\vartheta_{1},\vartheta_{2},\vartheta_{3})=\cos(\vartheta_{1}+\vartheta_{2})+\cos(\vartheta_{3})$,
then $n=3$ is not minimal, as it can be reduced to $n=2$ by taking
$\tilde{w}(\vartheta_{1},\vartheta_{2})=\cos(\vartheta_{1})+\cos(\vartheta_{2})$,
$\tilde{\Omega}=(\tilde{\Omega}_{1},\tilde{\Omega}_{2})\in{\mathbb{R}}^{2}$,
with $\tilde{\Omega}_{1}=\Omega_{1}+\Omega_{2}$ and
$\tilde{\Omega}_{2}=\Omega_{3}$, while $n=2$ cannot be further reduced. Hence
the function
$v(t)=\tilde{w}(\tilde{\Omega}_{1}t,\tilde{\Omega}_{2}t)=\cos(\tilde{\Omega}_{1}t)+\cos(\tilde{\Omega}_{2}t)$
is quasi-periodic with frequency vector $\tilde{\Omega}\in{\mathbb{R}}^{2}$.

For $n=2$, a vector $\Omega=(\Omega_{1},\Omega_{2})\in{\mathbb{R}}^{2}$ is
rationally independent if and only if $\Omega_{1}$ is nonzero and the ratio
$\Omega_{2}/\Omega_{1}$ is irrational. Hence a function
$v(t)=w(\Omega_{1}t,\Omega_{2}t)$ is quasi-periodic with frequency vector
$\Omega=(\Omega_{1},\Omega_{2})$ if $w(\vartheta_{1},\vartheta_{2})$ is
$2\pi$-periodic in $\vartheta_{1}$ and in $\vartheta_{2}$, $\Omega_{1}$ is
nonzero, $\Omega_{2}/\Omega_{1}$ is irrational, and $v(t)$ is not a periodic
function.

The main result of this paper is the following description of Gavrilov’s fluid
particle dynamics.

######  Theorem 1.1.

There exist universal positive constants
$\delta_{0},\tau_{0},\varepsilon_{0},I^{*}$ with the following properties. Let

$\mathcal{C},\delta,\mathcal{N},U,P,\varepsilon,\omega,W,\tilde{U},\tilde{P},\mathcal{S},\tau,\mathcal{N}^{\prime},\mathcal{S}^{*}$

be the sets, constants, and functions defined in Section 1.1 for $R=1$, with
$\delta=\delta_{0}$, $\tau=\tau_{0}$, and $0<\varepsilon\leq\varepsilon_{0}$.

$(i)$ There exists an analytic diffeomorphism

$\Phi:{\mathbb{T}}\times{\mathbb{T}}\times(0,I^{*})\to\mathcal{S}^{*},$

where ${\mathbb{T}}:={\mathbb{R}}/2\pi\mathbb{Z}$, such that the change of
variable $(x,y,z)=\Phi(\sigma,\beta,I)$ transforms system (1.11) into a system
of the form

$\dot{\sigma}=\Omega_{1}(I),\quad\ \dot{\beta}=\Omega_{2}(I),\quad\
\dot{I}=0.$ (1.12)

As a consequence, the solution $(x(t),y(t),z(t))$ of the Cauchy problem (1.11)
with initial datum

$(x(0),y(0),z(0))=(x_{0},y_{0},z_{0})=\Phi(\sigma_{0},\beta_{0},I_{0})\in\mathcal{S}^{*}$
(1.13)

is the function

$(x(t),y(t),z(t))=\Phi(\sigma(t),\beta(t),I(t)),$ (1.14)

defined for all $t\in{\mathbb{R}}$, where

$\sigma(t)=\sigma_{0}+\Omega_{1}(I_{0})t,\quad\
\beta(t)=\beta_{0}+\Omega_{2}(I_{0})t,\quad\ I(t)=I_{0}.$ (1.15)

The angle variables $\sigma(t),\beta(t)\in{\mathbb{T}}$ rotate with constant
angular frequencies $\Omega_{1}(I_{0}),\Omega_{2}(I_{0})$ respectively, and
the variable $I(t)=I_{0}$ is constant in time.

$(ii)$ The first and third equations of the transformed system (1.12) form a
Hamiltonian system

$\dot{\sigma}=\partial_{I}\mathcal{H}(\sigma,I),\quad\dot{I}=-\partial_{\sigma}\mathcal{H}(\sigma,I)$
(1.16)

with Hamiltonian $\mathcal{H}(\sigma,I)=\mathcal{H}(I)$ independent of the
angle variable $\sigma$; hence (1.16) is a Hamiltonian system in angle-action
variables.

$(iii)$ The frequency $\Omega_{1}(I)$ is given by

$\Omega_{1}(I)=\omega(\mathcal{K}(I))\,\mathcal{K}^{\prime}(I)$

where $\mathcal{K}(I)$ is the restriction to the interval $(0,I^{*})$ of a
function defined and analytic in the interval $(-I^{*},I^{*})$, with Taylor
expansion

$\mathcal{K}(I)=I+\frac{1065}{1024}I^{3}+O(I^{4})$

around $I=0$, and strictly increasing in $(-I^{*},I^{*})$. The frequency
$\Omega_{2}(I)$ is given by

$\Omega_{2}(I)=\sqrt{I}\,\mathcal{R}(I)\Omega_{1}(I)$

where $\mathcal{R}(I)$ is the restriction to the interval $(0,I^{*})$ of a
function defined and analytic in the interval $(-I^{*},I^{*})$, with Taylor
expansion

$\mathcal{R}(I)=1+\frac{7}{4}I+O(I^{2})$

around $I=0$. If $\omega(\mathcal{K}(I))\neq 0$, then both $\Omega_{1}(I)$ and
$\Omega_{2}(I)$ are nonzero, with ratio

$\frac{\Omega_{2}(I)}{\Omega_{1}(I)}=\sqrt{I}\,\mathcal{R}(I).$

The function $I\mapsto\sqrt{I}\,\mathcal{R}(I)$ is strictly increasing and
analytic in $(0,I^{*})$. Therefore it is rational for infinitely many values
of $I$, and irrational for infinitely many other values of $I$. More
precisely, the set $\\{I\in(0,I^{*}):\sqrt{I}\,\mathcal{R}(I)\in\mathbb{Q}\\}$
is a countable set, while the set
$\\{I\in(0,I^{*}):\sqrt{I}\,\mathcal{R}(I)\notin\mathbb{Q}\\}$ has full
Lebesgue measure.

$(iv)$ For $\omega(\mathcal{K}(I_{0}))\neq 0$, the solution (1.14) of the
Cauchy problem (1.11), (1.13) is periodic in time if
$\sqrt{I_{0}}\,\mathcal{R}(I_{0})$ is rational, and it is quasi-periodic in
time with frequency vector $(\Omega_{1}(I_{0}),\Omega_{2}(I_{0}))$ if
$\sqrt{I_{0}}\,\mathcal{R}(I_{0})$ is irrational.

$(v)$ The map $\Phi(\sigma,\beta,I)$ admits a converging expansion in powers
of $\sqrt{I}$ around $I=0$; more precisely, there exists a map
$\Psi(\sigma,\beta,\mu)$, defined and analytic in
${\mathbb{T}}^{2}\times(-\mu_{0},\mu_{0})$, where $\mu_{0}=\sqrt{I^{*}}$, such
that $\Phi(\sigma,\beta,I)=\Psi(\sigma,\beta,\sqrt{I})$ for all
$(\sigma,\beta,I)\in{\mathbb{T}}^{2}\times(0,I^{*})$. The map
$\Phi(\sigma,\beta,I)$ has the form

$\Phi(\sigma,\beta,I)=\begin{pmatrix}\rho(\sigma,I)\cos(\beta+\eta(\sigma,I))\\\
\rho(\sigma,I)\sin(\beta+\eta(\sigma,I))\\\ \zeta(\sigma,I)\end{pmatrix}$

where the functions $\rho(\sigma,I),\eta(\sigma,I),\zeta(\sigma,I)$ have
expansion

$\rho(\sigma,I)=1+\sqrt{2I}\sin\sigma+O(I),\quad\ \eta(\sigma,I)=O(I),\quad\
\zeta(\sigma,I)=\sqrt{2I}\cos\sigma+O(I).$

$(vi)$ The action variable $I$ and the pressure $P$ in (1.4) are related by
the identity

$P(\Phi(\sigma,\beta,I))=\mathcal{K}(I)\quad\
\forall(\sigma,\beta,I)\in{\mathbb{T}}^{2}\times(0,I^{*}).$

The pressure level and the action are in bijective correspondence; thus, the
action is a reparametrization of the pressure. The frequencies
$\Omega_{1}(I),\Omega_{2}(I)$ could also be expressed in terms of the pressure
level. The pressure $\tilde{P}$ in (1.6) satisfies
$\tilde{P}(\Phi(\sigma,\beta,I))=W(\mathcal{K}(I))$.

$(vii)$ The trajectory $\\{(x(t),y(t),z(t)):t\in{\mathbb{R}}\\}$ of the
solution (1.14) lies in the level set

$\mathcal{T}_{\ell}=\\{(x,y,z)\in{\mathbb{R}}^{3}:P(x,y,z)=\ell\\}$

of the pressure, where the value $\ell=P(x_{0},y_{0},z_{0})$ is determined by
the initial datum (1.13). The level set $\mathcal{T}_{\ell}$ has a nearly
toroidal shape, because $P(x,y,z)$ is given by (1.4), (1.5), and

$\alpha(\rho,z)=2(\rho-1)^{2}+2z^{2}+O((|\rho-1|+|z|)^{3})$

around $(\rho,z)=(1,0)$. More precisely, $\mathcal{T}_{\ell}$ is the
diffeomorphic image

$\mathcal{T}_{\ell}=\Phi({\mathbb{T}}^{2}\times\\{I\\})$

of the 2-dimensional torus
${\mathbb{T}}^{2}\times\\{I\\}=\\{(\sigma,\beta,I):(\sigma,\beta)\in{\mathbb{T}}^{2}\\}$,
where $I$ is determined by the identity

$\ell=\mathcal{K}(I).$ (1.17)

The pressure level $\ell$ determines whether the solution (1.14), (1.15) of
the Cauchy problem (1.11) with initial datum (1.13) on the surface
$\mathcal{T}_{\ell}$ is periodic or quasi-periodic, depending on the
rationality/irrationality of the ratio $\Omega_{2}(I)/\Omega_{1}(I)$, where
$I$ and $\ell$ are related by (1.17). Different solutions of system (1.11)
lying on the same surface $\mathcal{T}_{\ell}$, i.e., having the same pressure
level, share the same frequencies $\Omega_{1}(I),\Omega_{2}(I)$ and,
therefore, the same frequency ratio $\Omega_{2}(I)/\Omega_{1}(I)$.

######  Remark 1.2.

Theorem 1.1 is stated for cut-off functions $\omega$ supported in
$[\varepsilon,2\varepsilon]$, like those in [7]; however, Theorem 1.1, as well
as the result of [7], also holds for $\omega$ supported in any interval
$[\varepsilon_{1},\varepsilon_{2}]$ with
$0<\varepsilon_{1}<\varepsilon_{2}\leq\varepsilon_{0}$, without changing
anything in the proof. ∎

######  Remark 1.3.

By (1.16), system (1.12) is equivalent to the integrable Hamiltonian system
with two degrees of freedom in angle-action variables

$\dot{\sigma}=\partial_{I}\mathcal{H}^{+},\quad\dot{I}=-\partial_{\sigma}\mathcal{H}^{+},\quad\dot{\beta}=\partial_{K}\mathcal{H}^{+},\quad\dot{K}=-\partial_{\beta}\mathcal{H}^{+}$

with Hamiltonian
$\mathcal{H}^{+}(\sigma,I,\beta,K)=\mathcal{H}^{+}(I,K)=\mathcal{H}(I)+\Omega_{2}(I)K$,
restricted to the invariant set $K=0$. ∎

Theorem 1.1 is proved in Sections 3 and 4, splitting the proof into several
short simple steps. The proof uses only basic tools from the classical theory
of odes and dynamical systems.

###  1.3 Related literature

A general discussion about the existence of compactly supported smooth
solutions of pdes, also in comparison with the result of Gavrilov for the
steady Euler equations, can be found in the recent preprint [10]; in
particular, for Navier-Stokes equations, see [9].

The nice, explicit construction of Gavrilov’s original paper [7] has been
revisited and generalized by Constantin, La and Vicol in [4]. The paper [4]
also uses the result by Gavrilov as a building block to prove the existence of
compactly supported solutions of the steady Euler equations that have a given
Hölder regularity $C^{\alpha}$ and are not in $C^{\beta}$ for any
$\beta>\alpha$. The proof employs the invariances of the Euler equations and
the fact that the sum of compactly supported solutions with disjoint supports
is itself a solution.

The result by Gavrilov has also been used recently by Enciso, Peralta-Salas
and Torres de Lizaur in [6] to produce time-quasi-periodic solutions of the 3D
Euler equations. In [6] the authors extend to the 3D case, and to the
$n$-dimensional case for all $n$ even, the construction by Crouseilles and
Faou [5] of time-quasi-periodic solutions of the 2D Euler equations. Both [5]
and [6] use in a clever way the compactly supported solutions of the steady
equations as the main ingredients of the construction.

The time-quasi-periodic solutions in [5] and [6] are not of kam type, that is,
they are constructed outside the context of the Kolmogorov-Arnold-Moser
perturbation theory of nearly integrable dynamical systems, where small
divisor problems typically appear. Time-quasi-periodic solutions of the 3D
Euler equations of kam type are obtained in [3] by Montalto and the author in
presence of a forcing term, using pseudo-differential calculus and techniques
of kam theory for pdes.

By Theorem 1.1, the domain $\mathcal{S}^{*}$ is foliated by the 2-dimensional
tori $\mathcal{T}_{\ell}$, invariant for the vector field $\tilde{U}$, on
which the motion is periodic or quasi-periodic. This is true not only for the
Gavrilov vector field, but for all steady 3D Euler flows under suitable
assumptions, by a theorem of Arnold [1], [2]. Arnold’s theorem, and its role
in the study of the mixing property for Euler flows, is discussed by Khesin,
Kuksin and Peralta-Salas in [8].

##  2 Dual role of the parameter $R$ in Gavrilov’s solutions

Two different roles are played simultaneously by the parameter $R$ in the
Gavrilov’s solutions, because $R$ is

  * •

both a rescaling factor for the independent variable
$(x,y,z)\in{\mathbb{R}}^{3}$, appearing as $R^{-1}$ in the argument of
$\alpha$ in (1.5),

  * •

and an amplitude factor multiplying the vector fields $U,\tilde{U}$ and the
pressures $P,\tilde{P}$, appearing as powers $R^{3}$ and $R^{4}$ in the
definition of $b$ and $p$ in (1.5).

Separating these two different scalings helps to clarify the role of the
parameters in Gavrilov’s construction.

In fact, we observe that there exists a family of solutions described by two
real parameters $(\lambda,\mu)$ such that the solutions of Section 1.1 are
obtained in the special case $(\lambda,\mu)=(R,R^{2})$. This means that,
regarding the parameter $R$, Gavrilov’s solutions form a 1-parameter subset of
a 2-parameter family of solutions. Each element of the family is obtained from
any other element of the family by two simple rescalings, which correspond to
two basic invariances of the Euler equations, also preserving the
localizability condition. This allows us to study only one element of the
family (in particular, a normalized one), obtaining directly a description of
all the other elements.

Given $R>0$, let

$\mathcal{U}:=(\mathcal{C},\delta,\mathcal{N},U,P,\varepsilon,\omega,W,\tilde{U},\tilde{P},\mathcal{S},\tau,\mathcal{N}^{\prime},\mathcal{S}^{*})$
(2.1)

denote the list of the elements (sets, constants, and functions) defined in
Section 1.1. For every $\lambda>0$ and $\mu>0$, we define a rescaled version
of each element of the list $\mathcal{U}$ in the following way. We define

$\displaystyle\mathcal{A}_{\lambda,\mu}\mathcal{C}$
$\displaystyle:=\\{(x,y,z)\in{\mathbb{R}}^{3}:\rho=\lambda R,\ z=0\\},$
$\displaystyle\mathcal{A}_{\lambda,\mu}\delta$ $\displaystyle:=\lambda\delta,$
$\displaystyle\mathcal{A}_{\lambda,\mu}\mathcal{N}$
$\displaystyle:=\\{(x,y,z)\in{\mathbb{R}}^{3}:(\rho-\lambda
R)^{2}+z^{2}<(\lambda\delta)^{2}\\},$
$\displaystyle(\mathcal{A}_{\lambda,\mu}U)(x,y,z)$ $\displaystyle:=\mu
U\Big{(}\frac{x}{\lambda},\frac{y}{\lambda},\frac{z}{\lambda}\Big{)}\quad\
\forall(x,y,z)\in\mathcal{A}_{\lambda,\mu}\mathcal{N},$
$\displaystyle(\mathcal{A}_{\lambda,\mu}P)(x,y,z)$
$\displaystyle:=\mu^{2}P\Big{(}\frac{x}{\lambda},\frac{y}{\lambda},\frac{z}{\lambda}\Big{)}\quad\
\forall(x,y,z)\in\mathcal{A}_{\lambda,\mu}\mathcal{N},$
$\displaystyle\mathcal{A}_{\lambda,\mu}\varepsilon$
$\displaystyle:=\mu^{2}\varepsilon,$
$\displaystyle(\mathcal{A}_{\lambda,\mu}\omega)(s)$
$\displaystyle:=\omega\Big{(}\frac{s}{\mu^{2}}\Big{)}\quad\ \forall
s\in{\mathbb{R}},$ $\displaystyle(\mathcal{A}_{\lambda,\mu}W)(s)$
$\displaystyle:=\mu^{2}W\Big{(}\frac{s}{\mu^{2}}\Big{)}\quad\ \forall
s\in{\mathbb{R}},$ $\displaystyle(\mathcal{A}_{\lambda,\mu}\tilde{U})(x,y,z)$
$\displaystyle:=\mu\tilde{U}\Big{(}\frac{x}{\lambda},\frac{y}{\lambda},\frac{z}{\lambda}\Big{)}\quad\
\forall(x,y,z)\in{\mathbb{R}}^{3},$
$\displaystyle(\mathcal{A}_{\lambda,\mu}\tilde{P})(x,y,z)$
$\displaystyle:=\mu^{2}\tilde{P}\Big{(}\frac{x}{\lambda},\frac{y}{\lambda},\frac{z}{\lambda}\Big{)},\quad\
\forall(x,y,z)\in{\mathbb{R}}^{3},$
$\displaystyle\mathcal{A}_{\lambda,\mu}\mathcal{S}$
$\displaystyle:=\\{(x,y,z)\in\mathcal{A}_{\lambda,\mu}\mathcal{N}:\mathcal{A}_{\lambda,\mu}\varepsilon<(\mathcal{A}_{\lambda,\mu}P)(x,y,z)<2\mathcal{A}_{\lambda,\mu}\varepsilon\\},$
$\displaystyle\mathcal{A}_{\lambda,\mu}\tau$ $\displaystyle:=\mu^{2}\tau,$
$\displaystyle\mathcal{A}_{\lambda,\mu}\mathcal{N}^{\prime}$
$\displaystyle:=\\{(x,y,z)\in{\mathbb{R}}^{3}:(\rho-\lambda
R)^{2}+z^{2}<(\lambda\delta/4)^{2}\\},$
$\displaystyle\mathcal{A}_{\lambda,\mu}\mathcal{S}^{*}$
$\displaystyle:=\\{(x,y,z)\in\mathcal{A}_{\lambda,\mu}\mathcal{N}:0<(\mathcal{A}_{\lambda,\mu}P)(x,y,z)<3\mathcal{A}_{\lambda,\mu}\varepsilon\\},$
(2.2)

where $\rho=\sqrt{x^{2}+y^{2}}$. We denote by
$\mathcal{A}_{\lambda,\mu}\mathcal{U}:=(\mathcal{A}_{\lambda,\mu}\mathcal{C},\ldots,\mathcal{A}_{\lambda,\mu}\mathcal{S}^{*})$
the list of the rescaled elements. The properties of $\mathcal{U}$ in Section
1.1 become the following properties for
$\mathcal{A}_{\lambda,\mu}\mathcal{U}$.

  * •

The constant $\mathcal{A}_{\lambda,\mu}\delta=\lambda\delta$ satisfies
$0<\lambda\delta<\lambda R$ and $\lambda\delta\leq\lambda r_{0}R$.

  * •

The rescaled pressure $\mathcal{A}_{\lambda,\mu}P$ is analytic in
$\mathcal{A}_{\lambda,\mu}\mathcal{N}$.

  * •

The rescaled vector field $\mathcal{A}_{\lambda,\mu}U$ is continuous in
$\mathcal{A}_{\lambda,\mu}\mathcal{N}$ and analytic in
$(\mathcal{A}_{\lambda,\mu}\mathcal{N})\setminus(\mathcal{A}_{\lambda,\mu}\mathcal{C})$.

  * •

The pair $(\mathcal{A}_{\lambda,\mu}U,\,\mathcal{A}_{\lambda,\mu}P)$ satisfies
the Euler equations and the localizability condition in
$(\mathcal{A}_{\lambda,\mu}\mathcal{N})\setminus(\mathcal{A}_{\lambda,\mu}\mathcal{C})$.

  * •

The function $\mathcal{A}_{\lambda,\mu}\omega$ is
$C^{\infty}({\mathbb{R}},{\mathbb{R}})$ with support contained in
$[\mathcal{A}_{\lambda,\mu}\varepsilon,2\mathcal{A}_{\lambda,\mu}\varepsilon]=[\mu^{2}\varepsilon,2\mu^{2}\varepsilon]$.

  * •

The function $\mathcal{A}_{\lambda,\mu}W$ satisfies

$(\mathcal{A}_{\lambda,\mu}W)(s)=\int_{0}^{s}(\mathcal{A}_{\lambda,\mu}\omega)^{2}(\sigma)\,d\sigma\quad\
\forall s\in{\mathbb{R}}.$

  * •

The vector field $\mathcal{A}_{\lambda,\mu}\tilde{U}$ satisfies

$\displaystyle(\mathcal{A}_{\lambda,\mu}\tilde{U})(x,y,z)$
$\displaystyle=\mu\omega\Big{(}P\Big{(}\frac{x}{\lambda},\frac{y}{\lambda},\frac{z}{\lambda}\Big{)}\Big{)}U\Big{(}\frac{x}{\lambda},\frac{y}{\lambda},\frac{z}{\lambda}\Big{)}$
$\displaystyle=(\mathcal{A}_{\lambda,\mu}\omega)\big{(}(\mathcal{A}_{\lambda,\mu}P)(x,y,z)\big{)}\cdot(\mathcal{A}_{\lambda,\mu}U)(x,y,z)\quad\
\forall(x,y,z)\in\mathcal{A}_{\lambda,\mu}\mathcal{N}$

and $(\mathcal{A}_{\lambda,\mu}\tilde{U})(x,y,z)=0$ for all
$(x,y,z)\in{\mathbb{R}}^{3}\setminus(\mathcal{A}_{\lambda,\mu}\mathcal{N})$.

  * •

The pressure $\mathcal{A}_{\lambda,\mu}\tilde{P}$ satisfies

$\displaystyle(\mathcal{A}_{\lambda,\mu}\tilde{P})(x,y,z)$
$\displaystyle=\mu^{2}W\Big{(}P\Big{(}\frac{x}{\lambda},\frac{y}{\lambda},\frac{z}{\lambda}\Big{)}\Big{)}=(\mathcal{A}_{\lambda,\mu}W)\big{(}(\mathcal{A}_{\lambda,\mu}P)(x,y,z)\big{)}\quad\
\forall(x,y,z)\in\mathcal{A}_{\lambda,\mu}\mathcal{N}$

and
$(\mathcal{A}_{\lambda,\mu}\tilde{P})(x,y,z)=\mu^{2}W(2\varepsilon)=(\mathcal{A}_{\lambda,\mu}W)(2\mathcal{A}_{\lambda,\mu}\varepsilon)$
for all
$(x,y,z)\in{\mathbb{R}}^{3}\setminus(\mathcal{A}_{\lambda,\mu}\mathcal{N})$.

  * •

Both $\mathcal{A}_{\lambda,\mu}\tilde{U}$ and
$\mathcal{A}_{\lambda,\mu}\tilde{P}$ are $C^{\infty}({\mathbb{R}}^{3})$ and
satisfy the Euler equations and the localizability condition in
${\mathbb{R}}^{3}$.

  * •

Both $\mathcal{A}_{\lambda,\mu}\tilde{U}$ and
$\nabla(\mathcal{A}_{\lambda,\mu}\tilde{P})$ vanish outside the bounded set
$\mathcal{A}_{\lambda,\mu}\mathcal{S}$.

  * •

One has $\mathcal{A}_{\lambda,\mu}P>\mathcal{A}_{\lambda,\mu}\tau$ in
$(\mathcal{A}_{\lambda,\mu}\mathcal{N})\setminus(\mathcal{A}_{\lambda,\mu}\mathcal{N}^{\prime})$.

  * •

One has $\mathcal{A}_{\lambda,\mu}\tau\geq
3\mathcal{A}_{\lambda,\mu}\varepsilon$.

Thus, we have obtained the 2-parameter family
$\\{\mathcal{A}_{\lambda,\mu}\mathcal{U}\\}_{\lambda,\mu}$. One has the group
property

$\mathcal{A}_{\lambda_{1},\mu_{1}}(\mathcal{A}_{\lambda_{2},\mu_{2}}\mathcal{U})=\mathcal{A}_{\lambda_{1}\lambda_{2},\,\mu_{1}\mu_{2}}\mathcal{U},\quad\
\mathcal{A}_{\lambda,\mu}(\mathcal{A}_{\frac{1}{\lambda},\frac{1}{\mu}}\mathcal{U})=\mathcal{A}_{1,1}\mathcal{U}=\mathcal{U}$
(2.3)

for all $\lambda_{1},\lambda_{2},\lambda,\mu_{1},\mu_{2},\mu\in(0,\infty)$.
The check of (2.3) is straightforward.

######  Lemma 2.1.

Given $R>0$, let $\mathcal{U}_{R}$ be the list (2.1) of the elements (sets,
constants, and functions) defined in Section 1.1. Then

$\mathcal{U}_{R}=\mathcal{A}_{R,R^{2}}\,\mathcal{U}_{1}$

where $\mathcal{U}_{1}$ is a list of elements constructed in Section 1.1 for
$R=1$, and $\mathcal{A}_{R,R^{2}}$ is the rescaling operator
$\mathcal{A}_{\lambda,\mu}$, defined in (2.2), at $\lambda=R$, $\mu=R^{2}$.

###### Proof.

Let $\mathcal{U}_{R}$ be the list (2.1) of elements constructed in Section 1.1
for a given $R>0$. We observe that the list
$\mathcal{A}_{\lambda,\mu}\mathcal{U}_{R}$ with $\lambda=1/R$ and
$\mu=1/R^{2}$ coincides with a list of elements that one obtains by choosing
$R=1$ in Section 1.1, which we call $\mathcal{U}_{1}$. The check is
elementary; for example, regarding the vector field $U$ and the pressure $P$,
by (1.4) and (1.5) one has

$\displaystyle\frac{u_{\rho}(R\rho,Rz)}{R^{2}}$
$\displaystyle=\frac{\partial_{z}\alpha(\rho,z)}{4\rho},\qquad$
$\displaystyle\frac{u_{\varphi}(R\rho,Rz)}{R^{2}}$
$\displaystyle=\frac{\sqrt{H(\alpha(\rho,z))}}{4\rho},$
$\displaystyle\frac{u_{z}(R\rho,Rz)}{R^{2}}$
$\displaystyle=-\frac{\partial_{\rho}\alpha(\rho,z)}{4\rho},\qquad$
$\displaystyle\frac{P(Rx,Ry,Rz)}{R^{4}}$
$\displaystyle=\frac{\alpha(\rho,z)}{4}.$

Then, by (2.3),
$\mathcal{U}_{R}=\mathcal{A}_{R,R^{2}}(\mathcal{A}_{\frac{1}{R},\frac{1}{R^{2}}}\,\mathcal{U}_{R})=\mathcal{A}_{R,R^{2}}\,\mathcal{U}_{1}$.
∎

Regarding the fluid particle system (1.11), the consequence of Lemma 2.1 is
the following lemma, whose proof is trivial.

######  Lemma 2.2.

Let $\tilde{U}_{R}=\mathcal{A}_{R,R^{2}}\tilde{U}_{1}$, where $\tilde{U}_{R}$
is given by Section 1.1 for some $R>0$ and $\tilde{U}_{1}$ is given by Section
1.1 for $R=1$. Then a function $(x(t),y(t),z(t))$ solves the fluid particle
system (1.11) with velocity field $\tilde{U}=\tilde{U}_{R}$ if and only if

$x(t)=Rx_{1}(Rt),\quad\ y(t)=Ry_{1}(Rt),\quad\ z(t)=Rz_{1}(Rt),$

where $(x_{1}(t),y_{1}(t),z_{1}(t))$ solves

$(\dot{x}_{1}(t),\dot{y}_{1}(t),\dot{z}_{1}(t))=\tilde{U}_{1}(x_{1}(t),y_{1}(t),z_{1}(t)).$

##  3 Conjugation to a linear flow on ${\mathbb{T}}^{2}$

In this section and in Section 4 we prove Theorem 1.1. Thus, assume that
$\mathcal{U}$ in (2.1) is given by Section 1.1 for $R=1$. Hence, in
particular,

$\displaystyle\mathcal{C}$
$\displaystyle=\\{(x,y,z)\in{\mathbb{R}}^{3}:\rho=1,\ z=0\\},$
$\displaystyle\mathcal{N}$
$\displaystyle=\\{(x,y,z)\in{\mathbb{R}}^{3}:(\rho-1)^{2}+z^{2}<\delta^{2}\\},$
$\displaystyle\mathcal{N}^{\prime}$
$\displaystyle=\\{(x,y,z)\in{\mathbb{R}}^{3}:(\rho-1)^{2}+z^{2}<(\delta/4)^{2}\\},$
(3.1)

where $\rho=\sqrt{x^{2}+y^{2}}$. The constant $\delta$ satisfies $0<\delta<1$
and $\delta\leq r_{0}$. The vector field $U$ and the pressure $P$ in
$\mathcal{N}$ are given by (1.4), (1.5), where $a(\rho,z)=\alpha(\rho,z)$ and

$p(\rho,z)=\frac{\alpha(\rho,z)}{4},\quad\
b(\rho,z)=\frac{1}{4}\sqrt{H(\alpha(\rho,z))}.$ (3.2)

The constants $\tau$ and $\varepsilon$ satisfy (1.9) and $\tau\geq
3\varepsilon$. The function $\omega$ is supported in
$[\varepsilon,2\varepsilon]$, and $W$ is in (1.7). The vector field
$\tilde{U}$ and the pressure $\tilde{P}$ are given by (1.6) in $\mathcal{N}$,
and $(\tilde{U},\tilde{P})=(0,W(2\varepsilon))$ in
${\mathbb{R}}^{3}\setminus\mathcal{N}$. The sets $\mathcal{S},\mathcal{S}^{*}$
are given by (1.8), (1.10).

Outside the set $\mathcal{S}$, the vector field $\tilde{U}$ is zero, and the
solutions of system (1.11) are constant in time. Hence, by the uniqueness
property of the solution of Cauchy problems, any solution of (1.11) that is in
$\mathcal{S}$ at some time $t_{0}$ remains in $\mathcal{S}$ for its entire
lifespan; in other words, $\mathcal{S}$ is an invariant set for (1.11).
Moreover $\mathcal{S}$ is bounded, and therefore, by basic ode theory, the
solutions of (1.11) in $\mathcal{S}$ are all global in time.

Trivially, any subset of ${\mathbb{R}}^{3}\setminus\mathcal{S}$ is also an
invariant set for (1.11). Hence any subset of ${\mathbb{R}}^{3}$ containing
$\mathcal{S}$ is invariant for (1.11). In particular, $\mathcal{S}^{*}$ and
$\mathcal{N}$ are invariant for (1.11).

###  3.1 Cilindrical coordinates

To study system (1.11) in $\mathcal{N}$, first of all we move on to
cilindrical coordinates, that is, we consider the diffeomorphism

$\displaystyle\Phi_{1}$ $\displaystyle:\mathcal{N}_{1}\to\mathcal{N},\quad\
\Phi_{1}(\rho,\varphi,z):=(\rho\cos\varphi,\,\rho\sin\varphi,\,z),$
$\displaystyle\mathcal{N}_{1}$
$\displaystyle:=\\{(\rho,\varphi,z)\in{\mathbb{R}}\times{\mathbb{T}}\times{\mathbb{R}}:(\rho-1)^{2}+z^{2}<\delta^{2}\\},$
(3.3)

and the change of variables $\tilde{v}=\Phi_{1}(v)$, with $\tilde{v}=(x,y,z)$,
$v=(\rho,\varphi,z)$, i.e., $x=\rho\cos\varphi$, $y=\rho\sin\varphi$. Using
cilindrical coordinates is a natural choice because the quantities
$u_{\rho},u_{\varphi},u_{z},p$ in (1.5) are already expressed in terms of
$\rho,z$. Now a function $\tilde{v}(t)=\Phi_{1}(v(t))$ solves (1.11) in
$\mathcal{N}$ if and only if $v(t)$ solves

$\dot{v}(t)=V_{1}(v(t))$ (3.4)

in $\mathcal{N}_{1}$, where $V_{1}$ is the vector field

$V_{1}(v):=(D\Phi_{1}(v))^{-1}\tilde{U}(\Phi_{1}(v)),\quad\
v=(\rho,\varphi,z)\in\mathcal{N}_{1}.$

The Jacobian matrix $D\Phi_{1}(v)$ and its inverse matrix are

$D\Phi_{1}(v)=\begin{bmatrix}\cos\varphi&-\rho\sin\varphi&0\\\
\sin\varphi&\rho\cos\varphi&0\\\ 0&0&1\end{bmatrix},\quad\
(D\Phi_{1}(v))^{-1}=\begin{bmatrix}\cos\varphi&\sin\varphi&0\\\
-\frac{1}{\rho}\sin\varphi&\frac{1}{\rho}\cos\varphi&0\\\ 0&0&1\end{bmatrix},$

the composition $\tilde{U}(\Phi_{1}(v))$ is

$\tilde{U}(\Phi_{1}(v))=\omega(p(\rho,z))U(\Phi_{1}(v)),\quad\
U(\Phi_{1}(v))=\begin{pmatrix}u_{\rho}(\rho,z)\cos\varphi-
u_{\varphi}(\rho,z)\sin\varphi\\\
u_{\rho}(\rho,z)\sin\varphi+u_{\varphi}(\rho,z)\cos\varphi\\\
u_{z}(\rho,z)\end{pmatrix},$

and therefore

$\displaystyle V_{1}(\rho,\varphi,z)$
$\displaystyle=\omega(p(\rho,z))\Big{(}u_{\rho}(\rho,z),\
\frac{u_{\varphi}(\rho,z)}{\rho},\ u_{z}(\rho,z)\Big{)}$
$\displaystyle=\Big{(}\frac{\omega(p(\rho,z))\partial_{z}p(\rho,z)}{\rho},\
\frac{\omega(p(\rho,z))b(\rho,z)}{\rho^{2}},\
-\frac{\omega(p(\rho,z))\partial_{\rho}p(\rho,z)}{\rho}\Big{)}.$ (3.5)

Since $\alpha$ and $H$ are the functions constructed and studied in [7], it is
convenient to express the other quantities in terms of them. Hence, we define

$\chi(s):=\frac{1}{4}\omega\Big{(}\frac{s}{4}\Big{)}\quad\ \forall
s\in{\mathbb{R}},$ (3.6)

and, recalling (3.2), we rewrite (3.5) as

$V_{1}(\rho,\varphi,z)=\chi(\alpha(\rho,z))\Big{(}\frac{\partial_{z}\alpha(\rho,z)}{\rho},\
\frac{\sqrt{H(\alpha(\rho,z))}}{\rho^{2}},\
-\frac{\partial_{\rho}\alpha(\rho,z)}{\rho}\Big{)}.$ (3.7)

The curve $\mathcal{C}$ and the sets $\mathcal{S}^{*},\mathcal{N}^{\prime}$
become

$\displaystyle\mathcal{C}_{1}$
$\displaystyle:=\Phi_{1}^{-1}(\mathcal{C})=\\{(1,\varphi,0):\varphi\in{\mathbb{T}}\\},$
$\displaystyle\mathcal{S}_{1}^{*}$
$\displaystyle:=\Phi_{1}^{-1}(\mathcal{S}^{*})=\\{(\rho,\varphi,z)\in\mathcal{N}_{1}:0<\alpha(\rho,z)<4\tau\\},$
$\displaystyle\mathcal{N}_{1}^{\prime}$
$\displaystyle:=\Phi_{1}^{-1}(\mathcal{N}^{\prime})=\\{(\rho,\varphi,z)\in\mathcal{N}_{1}:(\rho-1)^{2}+z^{2}<(\delta/4)^{2}\\}.$

By (1.9) and (3.2), one has

$\alpha(\rho,z)>4\tau\quad\
\forall(\rho,z)\in\mathcal{N}_{1}\setminus\mathcal{N}_{1}^{\prime}.$ (3.8)

Moreover, the map $\Phi_{1}$ is analytic in $\mathcal{N}_{1}$.

The vector field $U$ satisfies the localizability condition (1.2) in
$\mathcal{N}\setminus\mathcal{C}$, and therefore $\tilde{U}\cdot\nabla P=0$ in
$\mathcal{N}$. Hence the pressure $P$ is a prime integral of system (1.11) in
$\mathcal{N}$. In cilindrical coordinates, this means that $p(\rho,z)$, and
therefore $\alpha(\rho,z)$ too, are prime integrals of (3.4) in
$\mathcal{N}_{1}$. This can also be verified directly: by (3.4) and (3.7),

$\displaystyle\frac{d}{dt}\big{\\{}\alpha(\rho(t),z(t))\big{\\}}$
$\displaystyle=\partial_{\rho}\alpha(\rho,z)\dot{\rho}+\partial_{z}\alpha(\rho,z)\dot{z}=0.$

Hence every trajectory $\\{v(t)=(\rho(t),\varphi(t),z(t)):t\in{\mathbb{R}}\\}$
of system (3.4) in $\mathcal{N}_{1}$ lies in a level set

$\mathcal{P}_{c}:=\\{(\rho,\varphi,z)\in\mathcal{N}_{1}:\alpha(\rho,z)=c\\}.$
(3.9)

In particular, every trajectory of (3.4) in $\mathcal{S}_{1}^{*}$ lies in a
level set $\mathcal{P}_{c}$ with $0<c<4\tau$. In fact, the set
$\mathcal{S}_{1}^{*}$ is exactly the union of all the level sets
$\mathcal{P}_{c}$ with $c\in(0,4\tau)$.

###  3.2 Elimination of the factor $1/\rho$ and canonical Hamiltonian
structure

We want to remove the factor $1/\rho$ appearing the first and third component
of $V_{1}$ in (3.5). We consider the diffeomorphism

$\displaystyle\Phi_{2}$
$\displaystyle:\mathcal{N}_{2}\to\mathcal{N}_{1},\quad\
\Phi_{2}(\rho,\varphi,z)=\Big{(}\rho,\varphi,\frac{z}{\rho}\Big{)},$
$\displaystyle\mathcal{N}_{2}$
$\displaystyle:=\\{(\rho,\varphi,z)\in{\mathbb{R}}\times{\mathbb{T}}\times{\mathbb{R}}:(\rho-1)^{2}+z^{2}\rho^{-2}<\delta^{2}\\},$
(3.10)

and the change of variables $\tilde{v}=\Phi_{2}(v)$, with
$\tilde{v}=(\tilde{\rho},\tilde{\varphi},\tilde{z})$, $v=(\rho,\varphi,z)$. A
function $\tilde{v}(t)=\Phi_{2}(v(t))$ solves (3.4) in $\mathcal{N}_{1}$ if
and only if $v(t)$ solves

$\dot{v}(t)=V_{2}(v(t))$ (3.11)

in $\mathcal{N}_{2}$, where

$V_{2}(v):=(D\Phi_{2}(v))^{-1}V_{1}(\Phi_{2}(v)),\quad\
v=(\rho,\varphi,z)\in\mathcal{N}_{2}.$ (3.12)

The Jacobian matrix $D\Phi_{2}(v)$ and its inverse are

$D\Phi_{2}(v)=\begin{bmatrix}1&0&0\\\ 0&1&0\\\
-z\rho^{-2}&0&\rho^{-1}\end{bmatrix},\quad\
(D\Phi_{2}(v))^{-1}=\begin{bmatrix}1&0&0\\\ 0&1&0\\\
z\rho^{-1}&0&\rho\end{bmatrix}.$

We define

$\alpha_{2}(\rho,z):=\alpha\Big{(}\rho,\frac{z}{\rho}\Big{)}$ (3.13)

for all $(\rho,z)\in\mathcal{D}_{2}$. Hence

$\partial_{z}\alpha_{2}(\rho,z)=(\partial_{z}\alpha)\Big{(}\rho,\frac{z}{\rho}\Big{)}\frac{1}{\rho},\quad\
\partial_{\rho}\alpha_{2}(\rho,z)=(\partial_{\rho}\alpha)\Big{(}\rho,\frac{z}{\rho}\Big{)}-(\partial_{z}\alpha)\Big{(}\rho,\frac{z}{\rho}\Big{)}\frac{z}{\rho^{2}},$

and, recalling the second identity in (3.2),

$V_{1}(\Phi_{2}(\rho,\varphi,z))=\chi(\alpha_{2}(\rho,z))\Big{(}\partial_{z}\alpha_{2}(\rho,z),\,\frac{\sqrt{H(\alpha_{2}(\rho,z))}}{\rho^{2}},\,-\frac{\partial_{\rho}\alpha_{2}(\rho,z)}{\rho}-\frac{z\partial_{z}\alpha_{2}(\rho,z)}{\rho^{2}}\Big{)}.$

Then the vector field $V_{2}$ in (3.12) is

$V_{2}(\rho,\varphi,z)=\chi(\alpha_{2}(\rho,z))\Big{(}\partial_{z}\alpha_{2}(\rho,z),\,\frac{\sqrt{H(\alpha_{2}(\rho,z))}}{\rho^{2}},\,-\partial_{\rho}\alpha_{2}(\rho,z)\Big{)}.$

The sets $\mathcal{C}_{1},\mathcal{S}_{1}^{*},\mathcal{N}_{1}^{\prime}$ become

$\displaystyle\mathcal{C}_{2}$
$\displaystyle:=\Phi_{2}^{-1}(\mathcal{C}_{1})=\\{(1,\varphi,0):\varphi\in{\mathbb{T}}\\}=\mathcal{C}_{1},$
$\displaystyle\mathcal{S}_{2}^{*}$
$\displaystyle:=\Phi_{2}^{-1}(\mathcal{S}_{1}^{*})=\\{(\rho,\varphi,z)\in\mathcal{N}_{2}:0<\alpha_{2}(\rho,z)<4\tau\\},$
$\displaystyle\mathcal{N}_{2}^{\prime}$
$\displaystyle:=\Phi_{2}^{-1}(\mathcal{N}_{1}^{\prime})=\\{(\rho,\varphi,z)\in\mathcal{N}_{2}:(\rho-1)^{2}+z^{2}\rho^{-2}<(\delta/4)^{2}\\}.$

Note that $\Phi_{2}$ leaves $\mathcal{C}_{2}=\mathcal{C}_{1}$ invariant,
because $z/\rho=z$ at $\rho=1$. By (3.8), one has

$\alpha_{2}(\rho,z)>4\tau\quad\
\forall(\rho,\varphi,z)\in\mathcal{N}_{2}\setminus\mathcal{N}_{2}^{\prime}.$
(3.14)

The level set $\mathcal{P}_{c}$ of $\alpha$ in (3.9) becomes the level set

$\mathcal{P}_{2,c}=\\{(\rho,\varphi,z)\in\mathcal{N}_{2}:\alpha_{2}(\rho,z)=c\\}$
(3.15)

of $\alpha_{2}$. The set $\mathcal{S}_{2}^{*}$ is the union of the level sets
$\mathcal{P}_{2,c}$ with $c\in(0,4\tau)$. The map $\Phi_{2}$ is analytic in
$\mathcal{N}_{2}$. The function $\alpha_{2}(\rho,z)$ is a prime integral of
system (3.11), and it is also analytic; its Taylor expansion around $(1,0)$ is
in (4.14). The first and third equation of system (3.11) are the Hamiltonian
system

$\dot{\rho}=\partial_{z}\mathcal{H}_{2}(\rho,z),\quad\
\dot{z}=-\partial_{\rho}\mathcal{H}_{2}(\rho,z),$ (3.16)

where

$\mathcal{H}_{2}(\rho,z):=\Gamma(\alpha_{2}(\rho,z)),\quad\
\Gamma(s):=\int_{0}^{s}\chi(\sigma)\,d\sigma.$ (3.17)

###  3.3 Symplectic polar coordinates in the radial-vertical plane

Now we move on to polar coordinates (in their symplectic version) to describe
the pair $(\rho,z)$. The pairs $(\rho,z)$ such that
$(\rho,\varphi,z)\in\mathcal{N}_{2}$ do not form a disc; thus, it is
convenient to consider a subset of $\mathcal{N}_{2}$ that fits with polar
coordinates better than how $\mathcal{N}_{2}$ does. We consider the open sets

$\displaystyle\mathcal{B}_{2}$
$\displaystyle:=\\{(\rho,\varphi,z)\in{\mathbb{R}}\times{\mathbb{T}}\times{\mathbb{R}}:0<(\rho-1)^{2}+z^{2}<\delta_{2}^{2}\\},$
$\displaystyle\mathcal{B}_{2}^{\prime}$
$\displaystyle:=\\{(\rho,\varphi,z)\in{\mathbb{R}}\times{\mathbb{T}}\times{\mathbb{R}}:0<(\rho-1)^{2}+z^{2}<(\delta_{2}/2)^{2}\\},$
(3.18)

where $\delta_{2}=C\delta$, $C>0$, and we observe that

$(\mathcal{B}_{2}\setminus\mathcal{B}_{2}^{\prime})\subseteq(\mathcal{N}_{2}\setminus\mathcal{N}_{2}^{\prime})$
(3.19)

if $C(1+\delta)\leq 1$ and $1+(\delta/4)\leq 2C$. This holds, for example, for
$C=2/3$ and all $0<\delta\leq 1/2$.

###### Proof of (3.19).

Let $(\rho,\varphi,z)\in\mathcal{B}_{2}\setminus\mathcal{B}_{2}^{\prime}$,
where $\delta_{2}=C\delta$, with $C(1+\delta)\leq 1$ and $1+(\delta/4)\leq
2C$. Since $(\rho,\varphi,z)\in\mathcal{B}_{2}$, one has
$(\rho-1)^{2}<\delta_{2}^{2}$, whence $\rho>1-\delta_{2}$. Moreover
$1-\delta_{2}=1-C\delta>0$ because $C\delta\leq 1-C<1$. Hence

$(\rho-1)^{2}+\frac{z^{2}}{\rho^{2}}\leq(\rho-1)^{2}+\frac{z^{2}}{(1-\delta_{2})^{2}}\leq\frac{(\rho-1)^{2}+z^{2}}{(1-\delta_{2})^{2}}<\frac{\delta_{2}^{2}}{(1-\delta_{2})^{2}}\leq\delta^{2},$

where the last inequality holds because $C(1+\delta)\leq 1$. Hence
$(\rho,\varphi,z)\in\mathcal{N}_{2}$. Now assume, by contradiction, that
$(\rho,\varphi,z)\in\mathcal{N}_{2}^{\prime}$. Then
$(\rho-1)^{2}<(\delta/4)^{2}$, whence $\rho<1+(\delta/4)$, and

$(\rho-1)^{2}+z^{2}\leq\Big{(}(\rho-1)^{2}+\frac{z^{2}}{\rho^{2}}\Big{)}\Big{(}1+\frac{\delta}{4}\Big{)}^{2}<\frac{\delta^{2}}{16}\Big{(}1+\frac{\delta}{4}\Big{)}^{2}\leq\frac{\delta_{2}^{2}}{4},$

where the last inequality holds because $1+(\delta/4)\leq 2C$. Also,
$0<(\rho-1)^{2}+z^{2}$ because $(\rho,\varphi,z)\in\mathcal{B}_{2}$. Therefore
$(\rho,\varphi,z)\in\mathcal{B}_{2}^{\prime}$, a contradiction. This proves
that $(\rho,\varphi,z)\notin\mathcal{N}_{2}^{\prime}$. ∎

By (3.19) and (3.14), one has

$\alpha_{2}(\rho,z)>4\tau\quad\
\forall(\rho,\varphi,z)\in\mathcal{B}_{2}\setminus\mathcal{B}_{2}^{\prime}.$
(3.20)

Thus
$\mathcal{S}_{2}^{*}\subseteq\mathcal{B}_{2}^{\prime}\subseteq\mathcal{B}_{2}$,
and they are invariant sets for system (3.11).

We consider the diffeomorphism

$\displaystyle\Phi_{3}:\mathcal{B}_{3}\to\mathcal{B}_{2},\quad\
\Phi_{3}(\vartheta,\varphi,\xi)=(1+\sqrt{2\xi}\sin\vartheta,\,\varphi,\,\sqrt{2\xi}\cos\vartheta),\quad\
\mathcal{B}_{3}:={\mathbb{T}}\times{\mathbb{T}}\times(0,\xi_{3}),\quad$ (3.21)

where $\xi_{3}:=\delta_{2}^{2}/2=2\delta^{2}/9$, and the change of variables
$\tilde{v}=\Phi_{3}(v)$, with
$\tilde{v}=(\tilde{\rho},\tilde{\varphi},\tilde{z})$,
$v=(\vartheta,\varphi,\xi)$. A function $\tilde{v}(t)=\Phi_{3}(v(t))$ solves
(3.11) in $\mathcal{B}_{2}$ if and only if $v(t)$ solves

$\dot{v}(t)=V_{3}(v(t))$ (3.22)

in $\mathcal{B}_{3}$, where

$V_{3}(v):=(D\Phi_{3}(v))^{-1}V_{2}(\Phi_{3}(v)),\quad\
v=(\vartheta,\varphi,\xi)\in\mathcal{B}_{3}.$

The Jacobian matrix $D\Phi_{3}(v)$ and its inverse are

$D\Phi_{3}(v)=\begin{bmatrix}\sqrt{2\xi}\cos\vartheta&0&\frac{1}{\sqrt{2\xi}}\sin\vartheta\\\
0&1&0\\\
-\sqrt{2\xi}\sin\vartheta&0&\frac{1}{\sqrt{2\xi}}\cos\vartheta\end{bmatrix},\quad(D\Phi_{3}(v))^{-1}=\begin{bmatrix}\frac{1}{\sqrt{2\xi}}\cos\vartheta&0&-\frac{1}{\sqrt{2\xi}}\sin\vartheta\\\
0&1&0\\\ \sqrt{2\xi}\sin\vartheta&0&\sqrt{2\xi}\cos\vartheta\end{bmatrix}.$

We define

$\displaystyle\alpha_{3}(\vartheta,\xi)$
$\displaystyle:=\alpha_{2}\big{(}1+\sqrt{2\xi}\sin\vartheta,\sqrt{2\xi}\cos\vartheta\big{)},$
(3.23)

for all $(\vartheta,\xi)\in{\mathbb{T}}\times[0,\xi_{3})$. In fact, in the set
$\mathcal{B}_{3}$, $\xi$ varies in the interval $(0,\xi_{3})$, but it is
convenient to consider $\alpha_{3}$ also for $\xi=0$; one has
$\alpha_{3}(\vartheta,0)=\alpha_{2}(1,0)=\alpha(1,0)=0$ (see (3.13)). From
(3.23) it follows that

$\displaystyle\partial_{\vartheta}\alpha_{3}(\vartheta,\xi)$
$\displaystyle=(\partial_{\rho}\alpha_{2})(\rho,z)\sqrt{2\xi}\cos\vartheta-(\partial_{z}\alpha_{2})(\rho,z)\sqrt{2\xi}\sin\vartheta,$
$\displaystyle\partial_{\xi}\alpha_{3}(\vartheta,\xi)$
$\displaystyle=(\partial_{\rho}\alpha_{2})(\rho,z)\frac{1}{\sqrt{2\xi}}\sin\vartheta+(\partial_{z}\alpha_{2})(\rho,z)\frac{1}{\sqrt{2\xi}}\cos\vartheta$
(3.24)

for all $(\vartheta,\xi)\in{\mathbb{T}}\times(0,\xi_{3})$, where
$(\rho,z)=(1+\sqrt{2\xi}\sin\vartheta,\sqrt{2\xi}\cos\vartheta)$. Hence

$V_{3}(\vartheta,\varphi,\xi)=\chi(\alpha_{3}(\vartheta,\xi))\Big{(}\partial_{\xi}\alpha_{3}(\vartheta,\xi),\
\frac{\sqrt{H(\alpha_{3}(\vartheta,\xi))}}{(1+\sqrt{2\xi}\sin\vartheta)^{2}},\
-\partial_{\vartheta}\alpha_{3}(\vartheta,\xi)\Big{)}.$ (3.25)

The set $\mathcal{C}_{2}$ is out of $\mathcal{B}_{2}$; the sets
$\mathcal{S}_{2}^{*}$, $\mathcal{B}_{2}^{\prime}$ become

$\displaystyle\mathcal{S}_{3}^{*}$
$\displaystyle:=\Phi_{3}^{-1}(\mathcal{S}_{2}^{*})=\\{(\vartheta,\varphi,\xi)\in\mathcal{B}_{3}:0<\alpha_{3}(\vartheta,\xi)<4\tau\\},$
$\displaystyle\mathcal{B}_{3}^{\prime}$
$\displaystyle:=\Phi_{3}^{-1}(\mathcal{B}_{2}^{\prime})={\mathbb{T}}\times{\mathbb{T}}\times(0,\xi_{3}/4).$

By (3.20), one has

$\alpha_{3}(\vartheta,\xi)>4\tau\quad\
\forall(\vartheta,\varphi,\xi)\in\mathcal{B}_{3}\setminus\mathcal{B}_{3}^{\prime}={\mathbb{T}}\times{\mathbb{T}}\times[\xi_{3}/4,\xi_{3}).$
(3.26)

Thus $\mathcal{S}_{3}^{*}\subset\mathcal{B}_{3}^{\prime}$. The level set
$\mathcal{P}_{2,c}$ of $\alpha_{2}$ in (3.15) becomes the level set

$\mathcal{P}_{3,c}=\\{(\vartheta,\varphi,\xi)\in\mathcal{B}_{3}:\alpha_{3}(\vartheta,\xi)=c\\}$

of $\alpha_{3}$. The set $\mathcal{S}_{3}^{*}$ is the union of the level sets
$\mathcal{P}_{3,c}$ with $c\in(0,4\tau)$. The map $\Phi_{3}$ is analytic in
$\mathcal{B}_{3}$. The function $\alpha_{3}(\vartheta,\xi)$ is a prime
integral of system (3.22); its behavior near $\xi=0$ is studied in Sections
3.4 and 4.3. The first and third equation of system (3.22) are the Hamiltonian
system

$\dot{\vartheta}=\partial_{\xi}\mathcal{H}_{3}(\vartheta,\xi),\quad\
\dot{\xi}=-\partial_{\vartheta}\mathcal{H}_{3}(\vartheta,\xi),$ (3.27)

where

$\mathcal{H}_{3}(\vartheta,\xi):=\mathcal{H}_{2}(1+\sqrt{2\xi}\sin\vartheta,\sqrt{2\xi}\cos\vartheta)=\Gamma(\alpha_{3}(\vartheta,\xi)),$
(3.28)

where $\mathcal{H}_{2}$ and $\Gamma$ are defined in (3.17). Moreover
$\vartheta$ (as well as $\varphi$) is an angle variable, i.e., it varies in
${\mathbb{T}}={\mathbb{R}}/2\pi\mathbb{Z}$.

######  Remark 3.1.

$(i)$ We use the symplectic transformation
$(\rho,z)=(1+\sqrt{2\xi}\sin\vartheta,\sqrt{2\xi}\cos\vartheta)$, instead of
the simpler polar coordinates $(\rho,z)=(1+r\cos\vartheta,r\sin\vartheta)$ or
$(\rho,z)=(1+r\sin\vartheta,r\cos\vartheta)$, in order to preserve the
canonical Hamiltonian structure of system (3.16).

$(ii)$ Using $\cos\vartheta$ for $z$ and $\sin\vartheta$ for $\rho$ in the
definition (3.21) of $\Phi_{3}$, instead of vice versa, is just a matter of
convenience: in this way, we get a positive angular velocity for the solution
$\vartheta(t)$. ∎

###  3.4 Level sets

Since the square root function $\xi\mapsto\sqrt{\xi}$ is analytic in
$(0,\infty)$, the function $\alpha_{3}(\vartheta,\xi)$ defined in (3.23) is
analytic in $(\vartheta,\xi)\in{\mathbb{T}}\times(0,\xi_{3})$. By the Taylor
expansion (4.14) of $\alpha_{2}(\rho,z)$, by (3.23) and (3.24), one has

$\displaystyle\alpha_{3}(\vartheta,\xi)$
$\displaystyle=4\xi+O(\xi^{\frac{3}{2}}),\quad\
\partial_{\xi}\alpha_{3}(\vartheta,\xi)=4+O(\xi^{\frac{1}{2}})\quad\ \text{as
}\xi\to 0,\ \xi>0,$ (3.29)

uniformly in $\vartheta\in{\mathbb{T}}$. Note that $\alpha_{3}(\vartheta,\xi)$
is not analytic around $\xi=0$, namely $\alpha_{3}(\vartheta,\xi)$, as a
function of $\xi$, is not a power series of $\xi$ centered at zero; in fact,
(3.29) is deduced from the analyticity of $\alpha_{2}(\rho,z)$ (and of its
partial derivatives $\partial_{\rho}\alpha_{2}(\rho,z)$,
$\partial_{z}\alpha_{2}(\rho,z)$) around $(1,0)$, and then from the evaluation
at $(\rho,z)=(1+\sqrt{2\xi}\sin\vartheta,\sqrt{2\xi}\cos\vartheta)$. See
Section 4.3 for more details.

Moreover, $\alpha_{3}(\vartheta,0)=\alpha_{2}(1,0)=0$, and, by (3.29),

$\partial_{\xi}\alpha_{3}(\vartheta,0)=\lim_{\xi\to
0^{+}}\frac{\alpha_{3}(\vartheta,\xi)-\alpha_{3}(\vartheta,0)}{\xi}=4=\lim_{\xi\to
0^{+}}\partial_{\xi}\alpha_{3}(\vartheta,\xi).$ (3.30)

Thus $\alpha_{3}$ is $C^{1}$ in ${\mathbb{T}}\times[0,\xi_{3})$. Taking
$\xi_{3}$ smaller (i.e., $\delta$ smaller) if necessary, we have
$\partial_{\xi}\alpha_{3}(\vartheta,\xi)>0$ for all
$(\vartheta,\xi)\in{\mathbb{T}}\times[0,\xi_{3})$. Hence the function
$\xi\mapsto\alpha_{3}(\vartheta,\xi)$ is strictly increasing on $[0,\xi_{3})$.
Moreover $\alpha_{3}(\vartheta,0)=0$ and, by (3.26),
$\alpha_{3}(\vartheta,\xi_{3}/4)>4\tau$ for all $\vartheta\in{\mathbb{T}}$.

As a consequence, for every $c\in[0,4\tau]$, for every
$\vartheta\in{\mathbb{T}}$, there exists a unique $\xi\in[0,\xi_{3}/4)$ such
that $\alpha_{3}(\vartheta,\xi)=c$. We denote $\gamma_{c}(\vartheta)$ the
unique solution $\xi$ of the equation $\alpha_{3}(\vartheta,\xi)=c$. Thus,

$\alpha_{3}(\vartheta,\gamma_{c}(\vartheta))=c\quad\
\forall(\vartheta,c)\in{\mathbb{T}}\times[0,4\tau].$ (3.31)

Moreover, $\gamma_{0}(\vartheta)=0$. Since $\alpha_{3}$ is analytic in
$(\vartheta,\xi)\in{\mathbb{T}}\times(0,\xi_{3})$, by the implicit function
theorem, the function $\gamma_{c}(\vartheta)$ is analytic in
$(\vartheta,c)\in{\mathbb{T}}\times(0,4\tau)$. The behavior of
$\gamma_{c}(\vartheta)$ around $c=0$ is studied in Section 4.3. From (3.31),

$\displaystyle(\partial_{\vartheta}\alpha_{3})(\vartheta,\gamma_{c}(\vartheta))+(\partial_{\xi}\alpha_{3})(\vartheta,\gamma_{c}(\vartheta))\partial_{\vartheta}\gamma_{c}(\vartheta)$
$\displaystyle=0,$ (3.32)
$\displaystyle(\partial_{\xi}\alpha_{3})(\vartheta,\gamma_{c}(\vartheta))\partial_{c}\gamma_{c}(\vartheta)$
$\displaystyle=1$ (3.33)

for all $(\vartheta,c)\in{\mathbb{T}}\times(0,4\tau)$. Since
$\partial_{\xi}\alpha_{3}(\vartheta,\xi)$ is positive for all
$(\vartheta,\xi)\in{\mathbb{T}}\times[0,\xi_{3})$, by (3.33) it follows that

$\partial_{c}\gamma_{c}(\vartheta)>0\quad\
\forall(\vartheta,c)\in{\mathbb{T}}\times(0,4\tau).$ (3.34)

Thus, for all $c\in[0,4\tau]$, the level set $\mathcal{P}_{3,c}$ is globally
described as the graph

$\mathcal{P}_{3,c}=\\{(\vartheta,\varphi,\xi)\in{\mathbb{T}}\times{\mathbb{T}}\times[0,\xi_{3}):\xi=\gamma_{c}(\vartheta)\\}=\\{(\vartheta,\varphi,\gamma_{c}(\vartheta)):(\vartheta,\varphi)\in{\mathbb{T}}^{2}\\}.$
(3.35)

###  3.5 Integration of the system in the radial-vertical plane

We consider the Cauchy problem of system (3.22) with initial datum
$(\vartheta_{0},\varphi_{0},\xi_{0})\in\mathcal{S}_{3}^{*}$ at the initial
time $t=0$. The components $\vartheta_{0}$ and $\xi_{0}$ of the initial datum
determine the level $c=\alpha_{3}(\vartheta_{0},\xi_{0})$; then the solution
$(\vartheta(t),\varphi(t),\xi(t))$ of the Cauchy problem is in the level set
$\mathcal{P}_{3,c}$ for all $t\in{\mathbb{R}}$, and, by (3.35),
$\xi(t)=\gamma_{c}(\vartheta(t))$ for all $t\in{\mathbb{R}}$.

The first and third equations of system (3.22) are the Hamiltonian system
(3.27), i.e.,

$\dot{\vartheta}=\chi(\alpha_{3}(\vartheta,\xi))\partial_{\xi}\alpha_{3}(\vartheta,\xi),\quad\
\dot{\xi}=-\chi(\alpha_{3}(\vartheta,\xi))\partial_{\vartheta}\alpha_{3}(\vartheta,\xi).$
(3.36)

Since $\xi(t)=\gamma_{c}(\vartheta(t))$, (3.36) becomes

$\dot{\vartheta}=\chi(c)(\partial_{\xi}\alpha_{3})(\vartheta,\gamma_{c}(\vartheta)),\quad\
\partial_{\vartheta}\gamma_{c}(\vartheta)\dot{\vartheta}=-\chi(c)(\partial_{\vartheta}\alpha_{3})(\vartheta,\gamma_{c}(\vartheta)).$
(3.37)

By (3.32), the second equation in (3.37) is equal to the first equation in
(3.37) multiplied by $\partial_{\vartheta}\gamma_{c}(\vartheta)$, and
therefore system (3.37) is equivalent to its first equation alone. Moreover,
by (3.33), the first equation in (3.37) is also

$\dot{\vartheta}=\frac{\chi(c)}{\partial_{c}\gamma_{c}(\vartheta)},$ (3.38)

where the denominator is nonzero by (3.33) or (3.34). Equation (3.38) with
initial datum $\vartheta(0)=\vartheta_{0}\in{\mathbb{T}}$ is an autonomous
Cauchy problem for the function $\vartheta(t)$ taking values in
${\mathbb{T}}$, and it can be integrated by basic calculus.

The function $\gamma_{c}(\vartheta)$ and its partial derivative
$\partial_{c}\gamma_{c}(\vartheta)$ are defined for
$\vartheta\in{\mathbb{T}}$, and hence they can also be considered as functions
defined for $\vartheta\in{\mathbb{R}}$ that are $2\pi$-periodic in
$\vartheta$. Thus, we first solve (3.38) considered as an equation for a
function $\vartheta^{r}(t)$ taking values in ${\mathbb{R}}$; then the
equivalence class of $\vartheta^{r}(t)$ mod $2\pi$ will be a function
$\vartheta(t)$ taking values in ${\mathbb{T}}$ and solving (3.38).

For all $c\in(0,4\tau)$, we define

$F_{c}:{\mathbb{R}}\to{\mathbb{R}},\quad\
F_{c}(\vartheta):=\int_{0}^{\vartheta}\partial_{c}\gamma_{c}(\sigma)\,d\sigma.$
(3.39)

By (3.34), $F_{c}$ is a diffeomorphism of ${\mathbb{R}}$. Let
$\vartheta^{r}_{0}\in{\mathbb{R}}$ be a representative of the equivalence
class $\vartheta_{0}\in{\mathbb{T}}$. If
$\vartheta^{r}:{\mathbb{R}}\to{\mathbb{R}}$, $t\mapsto\vartheta^{r}(t)$ solves
(3.38) with $\vartheta^{r}(0)=\vartheta_{0}^{r}$, then

$\frac{d}{dt}\\{F_{c}(\vartheta^{r}(t))\\}=F_{c}^{\prime}(\vartheta^{r}(t))\dot{\vartheta}^{r}(t)=\chi(c),\quad\
F_{c}(\vartheta^{r}(t))=F_{c}(\vartheta^{r}_{0})+\chi(c)t,$ (3.40)

and

$\vartheta^{r}(t)=F_{c}^{-1}\big{(}F_{c}(\vartheta^{r}_{0})+\chi(c)t\big{)}.$
(3.41)

Hence $\vartheta^{r}(t)$ in (3.41) is the unique solution of equation (3.38)
with initial datum $\vartheta^{r}(0)=\vartheta^{r}_{0}$. Now let
$\vartheta(t)$ be the equivalence class mod $2\pi$ of $\vartheta^{r}(t)$,
i.e.,

$\vartheta(t):=\\{\vartheta^{r}(t)+2k\pi:k\in\mathbb{Z}\\},$ (3.42)

for all $t\in{\mathbb{R}}$. Then the function $\vartheta(t)$ in (3.42) is a
function of time taking values in ${\mathbb{T}}$, and it solves (3.38) with
$\vartheta(0)=\vartheta_{0}$.

###  3.6 Rotation period in the radial-vertical plane

We show that the function $\vartheta(t)$ in (3.42) is periodic, and we
calculate its period. We begin with observing that the function $F_{c}$
defined in (3.39) satisfies

$F_{c}(2\pi+\vartheta)=F_{c}(2\pi)+F_{c}(\vartheta)\quad\
\forall\vartheta\in{\mathbb{R}}.$ (3.43)

###### Proof of (3.43).

Consider the definition (3.39), split the integral over $[0,2\pi+\vartheta]$
into the sum of $(i)$ the integral over $[0,2\pi]$ and $(ii)$ the integral
over $[2\pi,2\pi+\vartheta]$; then $(i)=F(2\pi)$ by definition, while
$(ii)=F(\vartheta)$ because the function $\partial_{c}\gamma(\sigma,c)$ is
$2\pi$-periodic in $\sigma$ and therefore $(ii)$ is equal to the integral over
$[0,\vartheta]$. ∎

Suppose that $\chi(c)$ is nonzero. Then the function $\vartheta^{r}(t)$
defined in (3.41) satisfies

$\vartheta^{r}(t+T_{c})=\vartheta^{r}(t)+2\pi\quad\forall
t\in{\mathbb{R}},\qquad\qquad T_{c}:=\frac{F_{c}(2\pi)}{\chi(c)}.$ (3.44)

###### Proof of (3.44).

Applying (3.40) twice, one has

$F_{c}(\vartheta^{r}(t+T_{c}))=F_{c}(\vartheta_{0}^{r})+\chi(c)(t+T_{c})=F_{c}(\vartheta^{r}(t))+\chi(c)T_{c}.$

By the definition of $T_{c}$ in (3.44) and by identity (3.43),

$F_{c}(\vartheta^{r}(t))+\chi(c)T_{c}=F_{c}(\vartheta^{r}(t))+F_{c}(2\pi)=F_{c}(\vartheta^{r}(t)+2\pi).$

Hence $F_{c}(\vartheta^{r}(t+T_{c}))=F_{c}(\vartheta^{r}(t)+2\pi)$, and, since
$F_{c}$ is invertible, we obtain (3.44). ∎

From (3.44) it follows that $\vartheta(t)$ defined in (3.42) is periodic in
time with period $T_{c}$.

###  3.7 Angle-action variables in the radial-vertical plane

The Hamiltonian system (3.27) has one degree of freedom, and therefore, by the
classical theory of Hamiltonian systems, it is completely integrable both in
the sense that the differential equations can be solved by quadrature (i.e.,
they can be transformed into a problem of finding primitives of functions),
and in the sense that they admit “angle-action variables” (this is the one-
dimensional, simplest case of the Liouville-Arnold theorem). System (3.27) has
been integrated in Section 3.5; now we construct its angle-action variables.

Given a point $(\vartheta^{*},\varphi^{*},\xi^{*})\in\mathcal{S}_{3}^{*}$, we
calculate its level $c=\alpha_{4}(\vartheta^{*},\xi^{*})$, which is in the
interval $(0,12\varepsilon)$, and we consider the Cauchy problem

$\dot{\vartheta}=\frac{1}{\partial_{c}\gamma_{c}(\vartheta)},\quad\
\vartheta(0)=0.$ (3.45)

The first equation in (3.45) is equation (3.38) in which $\chi(c)$ is replaced
by $1$. Hence, following Section 3.5 with 1 instead of $\chi(c)$, the
${\mathbb{R}}$-valued solution of (3.45) is given by (3.41) in which $\chi(c)$
is replaced by $1$ (note that the functions $\alpha_{3}(\vartheta,\xi)$,
$\gamma_{c}(\vartheta)$, $F_{c}(\vartheta)$ are all independent of $\chi(c)$).
We indicate $\vartheta^{r}_{c}(t)$ the ${\mathbb{R}}$-valued solution of
(3.45). Thus, since $F_{c}(0)=0$,

$\vartheta^{r}_{c}(t)=F_{c}^{-1}(t)\quad\ \forall t\in{\mathbb{R}}.$ (3.46)

Moreover, following Section 3.6 with 1 instead of $\chi(c)$, one has

$\vartheta^{r}_{c}(t+T_{c}^{*})=\vartheta^{r}_{c}(t)+2\pi\quad\forall
t\in{\mathbb{R}},\qquad\qquad T^{*}_{c}:=F_{c}(2\pi).$ (3.47)

We fix the representative $\vartheta_{1}^{*}$ of the class
$\vartheta^{*}\in{\mathbb{T}}$ such that $\vartheta^{*}_{1}\in[0,2\pi)$, and
we take the unique real number $s_{1}^{*}\in[0,T_{c}^{*})$ such that
$\vartheta^{r}_{c}(s_{1}^{*})=\vartheta^{*}_{1}$, i.e., by (3.46),
$s_{1}^{*}=F_{c}(\vartheta_{1}^{*})$. Then we define
$\sigma_{1}^{*}=s_{1}^{*}2\pi/T_{c}^{*}$, and we note that
$\sigma_{1}^{*}\in[0,2\pi)$ because $s_{1}^{*}\in[0,T_{c}^{*})$. By the
definition of $\sigma_{1}^{*},s_{1}^{*},T_{c}^{*}$, one has
$\sigma_{1}^{*}=f_{c}(\vartheta_{1}^{*})$, where

$f_{c}:{\mathbb{R}}\to{\mathbb{R}},\quad\
f_{c}(\vartheta):=\frac{F_{c}(\vartheta)2\pi}{F_{c}(2\pi)}\quad\forall\vartheta\in{\mathbb{R}}.$
(3.48)

Also, $\vartheta_{1}^{*}=\vartheta_{c}^{r}(s_{1}^{*})$ and
$s_{1}^{*}=\sigma_{1}^{*}T_{c}^{*}/2\pi$, whence
$\vartheta_{1}^{*}=g_{c}(\sigma_{1}^{*})$, where

$g_{c}:{\mathbb{R}}\to{\mathbb{R}},\quad\
g_{c}(\sigma):=\vartheta^{r}_{c}\Big{(}\frac{\sigma
T_{c}^{*}}{2\pi}\Big{)}\quad\forall\sigma\in{\mathbb{R}}.$ (3.49)

The function $f_{c}$ is a diffeomorphism of ${\mathbb{R}}$ because it is a
multiple of $F_{c}$, and it satisfies

$f_{c}(\vartheta+2\pi)=f_{c}(\vartheta)+2\pi\quad\
\forall\vartheta\in{\mathbb{R}}$ (3.50)

because $F_{c}$ satisfies (3.43). The function $g_{c}$ also satisfies

$g_{c}(\sigma+2\pi)=g_{c}(\sigma)+2\pi\quad\ \forall\sigma\in{\mathbb{R}}$
(3.51)

because $\vartheta^{r}_{c}$ satisfies (3.47). By (3.46), (3.48), (3.49),
$g_{c}(f_{c}(\vartheta))=\vartheta$ for all $\vartheta\in{\mathbb{R}}$ and
$f_{c}(g_{c}(\sigma))=\sigma$ for all $\sigma\in{\mathbb{R}}$, i.e., $g_{c}$
is the inverse diffeomorphism of $f_{c}$.

By (3.50) and (3.51), $f_{c}$ and $g_{c}$ induce diffeomorphisms of the torus

$f_{c}:{\mathbb{T}}\to{\mathbb{T}},\quad\
g_{c}:{\mathbb{T}}\to{\mathbb{T}},\quad\
g_{c}(f_{c}(\vartheta))=\vartheta\quad\forall\vartheta\in{\mathbb{T}},\quad\
f_{c}(g_{c}(\sigma))=\sigma\quad\forall\sigma\in{\mathbb{T}}$

(with a common little abuse, we use the same notation for the diffeomorphisms
of ${\mathbb{R}}$ and for the corresponding induced diffeomorphism of
${\mathbb{T}}$). Now $\sigma_{1}^{*}$ and $\vartheta_{1}^{*}$ are related by
the identities $\vartheta_{1}^{*}=g_{c}(\sigma_{1}^{*})$,
$\sigma_{1}^{*}=f_{c}(\vartheta_{1}^{*})$. By the definition of
$\vartheta_{1}^{*}$, $\vartheta^{*}\in{\mathbb{T}}$ is the equivalence class
of $\vartheta_{1}^{*}$ mod $2\pi$; let $\sigma^{*}\in{\mathbb{T}}$ be the
equivalence class of $\sigma_{1}^{*}$ mod $2\pi$. Then

$\vartheta^{*}=g_{c}(\sigma^{*}),\quad\ \sigma^{*}=f_{c}(\vartheta^{*}),\quad\
\vartheta^{*},\sigma^{*}\in{\mathbb{T}}.$

By (3.31), $\xi^{*}=\gamma_{c}(\vartheta^{*})$. Thus, we have expressed
$(\vartheta^{*},\xi^{*})$ in terms of $(\sigma^{*},c)$. From now on, we write
$\sigma$ instead of $\sigma^{*}$.

Now we introduce a variable $I$ and a function $h$ to express the level $c$ in
terms of $I$, i.e., $c=h(I)$; the function $h$ is a diffeomorphism (to be
determined) of some interval $(0,I^{*})$ (to be determined) onto the interval
$(0,4\tau)$ to which $c$ belongs. We consider the map

$\Phi_{4}(\sigma,\varphi,I):=(g_{c}(\sigma),\varphi,\gamma_{c}(g_{c}(\sigma)))|_{c=h(I)}=(g_{h(I)}(\sigma),\,\varphi,\,\gamma_{h(I)}(g_{h(I)}(\sigma))),$
(3.52)

defined on the set

$\mathcal{S}_{4}^{*}:=\\{(\sigma,\varphi,I):\sigma,\varphi\in{\mathbb{T}},\,I\in(0,I^{*})\\}={\mathbb{T}}\times{\mathbb{T}}\times(0,I^{*}),$
(3.53)

and we want to calculate its Jacobian determinant. One has

$\displaystyle\partial_{\sigma}\\{g_{c}(\sigma)\\}$
$\displaystyle=g_{c}^{\prime}(\sigma),\quad\
\partial_{I}\\{g_{c}(\sigma)\\}=\partial_{c}g_{c}(\sigma)h^{\prime}(I),\quad\
\partial_{\sigma}\\{\gamma_{c}(g_{c}(\sigma))\\}=\gamma_{c}^{\prime}(g_{c}(\sigma))g_{c}^{\prime}(\sigma),$
$\displaystyle\partial_{I}\\{\gamma_{c}(g_{c}(\sigma))\\}$
$\displaystyle=\big{\\{}(\partial_{c}\gamma_{c})(g_{c}(\sigma))+\gamma_{c}^{\prime}(g_{c}(\sigma))\partial_{c}g_{c}(\sigma))\big{\\}}h^{\prime}(I)$

where $c=h(I)$. Hence the Jacobian matrix is

$D\Phi_{4}(\sigma,\varphi,I)=\begin{bmatrix}g_{c}^{\prime}(\sigma)&0&\partial_{c}g_{c}(\sigma)h^{\prime}(I)\\\
0&1&0\\\
\gamma_{c}^{\prime}(g_{c}(\sigma))g_{c}^{\prime}(\sigma)&0&\big{\\{}(\partial_{c}\gamma_{c})(g_{c}(\sigma))+\gamma_{c}^{\prime}(g_{c}(\sigma))\partial_{c}g_{c}(\sigma))\big{\\}}h^{\prime}(I)\end{bmatrix},$

and its determinant is

$\det
D\Phi_{4}(\sigma,\varphi,I)=g_{c}^{\prime}(\sigma)\,(\partial_{c}\gamma_{c})(g_{c}(\sigma))\,h^{\prime}(I)$

where $c=h(I)$. By (3.49) and (3.45),

$g_{c}^{\prime}(\sigma)=\frac{1}{(\partial_{c}\gamma_{c})(g_{c}(\sigma))}\frac{T_{c}^{*}}{2\pi},$

and $T_{c}^{*}=F_{c}(2\pi)$, see (3.47). Hence

$\det D\Phi_{4}(\sigma,\varphi,I)=\frac{F_{c}(2\pi)}{2\pi}h^{\prime}(I)$
(3.54)

where $c=h(I)$. We want the determinant (3.54) to be $=1$, so that the
transformation $\Phi_{4}$ is symplectic and the Hamiltonian structure is
preserved. The average

$h_{1}(c):=\frac{1}{2\pi}\int_{0}^{2\pi}\gamma_{c}(\vartheta)\,d\vartheta$
(3.55)

is a strictly increasing function of $c\in[0,4\tau]$ because its derivative
$h_{1}^{\prime}(c)=F_{c}(2\pi)/2\pi$ is positive for all $c\in(0,4\tau)$ by
(3.34) and (3.39). Since $\gamma_{0}(\vartheta)=0$, one has $h_{1}(0)=0$. We
define

$I^{*}:=h_{1}(4\tau),$ (3.56)

and we note that, by the definition of $\gamma_{c}(\vartheta)$, one has
$0<I^{*}<\xi_{3}/4$. Thus, the interval $(0,I^{*})$ is the image
$\\{h_{1}(c):c\in(0,4\tau)\\}$ of the interval $(0,4\tau)$. We define
$h:(0,I^{*})\to(0,4\tau)$ as the inverse of $h_{1}:(0,4\tau)\to(0,I^{*})$.
Thus,

$h^{\prime}(I)>0\quad\ \forall I\in(0,I^{*}).$ (3.57)

This concludes the definition of the map $\Phi_{4}$ in (3.52), which is a
diffeomorphism of $\mathcal{S}_{4}^{*}$ onto $\mathcal{S}_{3}^{*}$.

Now that the transformation $\Phi_{4}$ has been defined, we use it to
transform system (3.22). We consider the change of variables
$\tilde{v}=\Phi_{4}(v)$, where
$\tilde{v}=(\vartheta,\varphi,\xi)\in\mathcal{S}_{3}^{*}$,
$v=(\sigma,\varphi,I)\in\mathcal{S}_{4}^{*}$. This means that
$\vartheta=g_{h(I)}(\sigma)$ and $\xi=\gamma_{h(I)}(g_{h(I)}(\sigma))$, that
is, $\vartheta=g_{c}(\sigma)$ and $\xi=\gamma_{c}(g_{c}(\sigma))$ where
$c=h(I)$. A function $\tilde{v}(t)=\Phi_{4}(v(t))$ solves (3.22) in
$\mathcal{S}_{3}^{*}$ if and only if $v(t)$ solves

$\dot{v}(t)=V_{4}(v(t))$ (3.58)

in $\mathcal{S}_{4}^{*}$, where

$V_{4}(v):=(D\Phi_{4}(v))^{-1}V_{3}(\Phi_{4}(v)),\quad\
v=(\sigma,\varphi,I)\in\mathcal{S}_{4}^{*}.$ (3.59)

The second row of the inverse matrix $(D\Phi_{4}(v))^{-1}$ is $(0,1,0)$, and
therefore the second component of the vector field $V_{4}$ is simply the
second component of $V_{3}$ (see (3.25)) evaluated at
$\Phi_{4}(\sigma,\varphi,I)$, which is $\chi(h(I))Q(\sigma,I)$, where

$Q(\sigma,I):=\frac{\sqrt{H(h(I))}}{\left(1+\sqrt{2\gamma_{h(I)}(g_{h(I)}(\sigma))}\sin(g_{h(I)}(\sigma))\right)^{2}}.$
(3.60)

Since the first and third equations of (3.58) are the Hamiltonian system
(3.27) and $\Phi_{4}$ is symplectic in the $(\sigma,I)$ variables, the first
and third components of the vector field $V_{4}$ are
$\partial_{I}\mathcal{H}_{4}(\sigma,I)$ and
$-\partial_{\sigma}\mathcal{H}_{4}(\sigma,I)$ respectively, where
$\mathcal{H}_{4}:=\mathcal{H}_{3}\circ\Phi_{4}$ (of course this can also be
checked directly, without using the properties of the symplectic
transformations). Now $\mathcal{H}_{3}$ is defined in (3.28), with $\Gamma$
defined in (3.17). Hence

$\displaystyle\mathcal{H}_{4}(\sigma,I)$
$\displaystyle=\mathcal{H}_{4}(I)=\Gamma(h(I)),\quad\
\partial_{I}\mathcal{H}_{4}(I)=\chi(h(I))h^{\prime}(I),$ (3.61)

and system (3.58) is

$\dot{\sigma}=\partial_{I}\mathcal{H}_{4}(I),\quad\
\dot{\varphi}=\chi(h(I))Q(\sigma,I),\quad\ \dot{I}=0$ (3.62)

in $\mathcal{S}_{4}^{*}$, with $Q(\sigma,I)$ defined in (3.60). The action $I$
is constant in time. The angle $\sigma$ rotates with constant angular velocity
$\partial_{I}\mathcal{H}_{4}(I)=\chi(h(I))h^{\prime}(I)$.

###  3.8 Reduction to a constant rotation in the tangential direction

Now that the equations of the motion in the radial-vertical plane have been
written in angle-action variables $(\sigma,I)$ in (3.62), we want to obtain a
similar simplification for the equation of the motion in the tangential
direction $\dot{\varphi}$. Recalling the definition (3.53) of
$\mathcal{S}_{4}^{*}$, we consider the diffeomorphism

$\Phi_{5}:\mathcal{S}_{4}^{*}\to\mathcal{S}_{4}^{*},\quad\
\Phi_{5}(\sigma,\beta,I)=(\sigma,\beta+\eta(\sigma,I),I),$ (3.63)

where $\eta:{\mathbb{T}}\times(0,I^{*})\to{\mathbb{R}}$ is a function to be
determined. We consider the change of variables $\tilde{v}=\Phi_{5}(v)$, where
$\tilde{v}=(\sigma,\varphi,I)\in\mathcal{S}_{4}^{*}$, $v=(\sigma,\beta,I)$
$\in\mathcal{S}_{4}^{*}$. A function $\tilde{v}(t)=\Phi_{5}(v(t))$ solves
(3.58) in $\mathcal{S}_{4}^{*}$ if and only if $v(t)$ solves

$\dot{v}(t)=V_{5}(v(t))$ (3.64)

in $\mathcal{S}_{4}^{*}$, where

$V_{5}(v):=(D\Phi_{5}(v))^{-1}V_{4}(\Phi_{5}(v)),\quad\
v=(\sigma,\beta,I)\in\mathcal{S}_{4}^{*}.$

The Jacobian matrix and its inverse are

$D\Phi_{5}(v)=\begin{bmatrix}1&0&0\\\
\partial_{\sigma}\eta&1&\partial_{I}\eta\\\ 0&0&1\end{bmatrix},\quad\
(D\Phi_{5}(v))^{-1}=\begin{bmatrix}1&0&0\\\
-\partial_{\sigma}\eta&1&-\partial_{I}\eta\\\ 0&0&1\end{bmatrix}.$

By (3.61) and (3.62), the vector field $V_{4}$ in (3.59) is

$V_{4}(\sigma,\varphi,I)=\chi(h(I))\,(h^{\prime}(I),\ Q(\sigma,I),\ 0).$

Hence

$V_{5}(\sigma,\beta,I)=\chi(h(I))\,(h^{\prime}(I),\
Q(\sigma,I)-\partial_{\sigma}\eta(\sigma,I)h^{\prime}(I),\ 0).$

We decompose $Q(\sigma,I)$ as the sum of its average in
$\sigma\in{\mathbb{T}}$ and its zero-average remainder, that is,

$Q(\sigma,I)=Q_{0}(I)+\tilde{Q}(\sigma,I),\quad\
Q_{0}(I):=\frac{1}{2\pi}\int_{0}^{2\pi}Q(\sigma,I)\,d\sigma,\quad\
\tilde{Q}:=Q-Q_{0},$ (3.65)

and define

$\eta(\sigma,I):=\frac{1}{h^{\prime}(I)}\int_{0}^{\sigma}\tilde{Q}(s,I)\,ds.$
(3.66)

Note that $h^{\prime}(I)$ is nonzero by (3.57). The function $\eta$ is
$2\pi$-periodic in $\sigma$ because $\tilde{Q}$ is $2\pi$-periodic in $\sigma$
with zero average on ${\mathbb{T}}$. By (3.66) and (3.65),

$\partial_{\sigma}\eta(\sigma,I)h^{\prime}(I)=\tilde{Q}(\sigma,I),\quad\
Q(\sigma,I)-\partial_{\sigma}\eta(\sigma,I)h^{\prime}(I)=Q_{0}(I).$

Thus, the vector field $V_{5}$ depends only on the action variable $I$, and it
is

$V_{5}(\sigma,\beta,I)=V_{5}(I)=(\Omega_{1}(I),\Omega_{2}(I),0),$

where

$\Omega_{1}(I):=\partial_{I}\mathcal{H}_{4}(I)=\chi(h(I))h^{\prime}(I),\quad\
\Omega_{2}(I):=\chi(h(I))Q_{0}(I).$ (3.67)

Then system (3.64) is

$\dot{\sigma}=\Omega_{1}(I),\quad\ \dot{\beta}=\Omega_{2}(I),\quad\
\dot{I}=0.$ (3.68)

###  3.9 Ratio of the two rotation periods

Let
$(\sigma_{0},\beta_{0},I_{0})\in\mathcal{S}_{4}^{*}={\mathbb{T}}^{2}\times(0,I^{*})$.
Let $\sigma_{0}^{r}\in{\mathbb{R}}$ be a representative of the equivalence
class $\sigma_{0}\in{\mathbb{T}}$, and let $\beta_{0}^{r}\in{\mathbb{R}}$ be a
representative of $\beta_{0}\in{\mathbb{T}}$. The solution of the Cauchy
problem for (3.68) in ${\mathbb{R}}^{2}\times(0,I^{*})$ with initial data
$(\sigma_{0}^{r},\beta_{0}^{r},I_{0})$ — i.e., system (3.68) where the first
two equations are considered as equations for functions $\sigma^{r}(t)$,
$\beta^{r}(t)$ taking values in ${\mathbb{R}}$ with initial data
$\sigma_{0}^{r},\beta_{0}^{r}$ — is

$\sigma^{r}(t)=\sigma_{0}^{r}+\Omega_{1}(I_{0})t,\quad\
\beta^{r}(t)=\beta_{0}^{r}+\Omega_{2}(I_{0})t,\quad\ I(t)=I_{0}\quad\ \forall
t\in{\mathbb{R}}.$ (3.69)

Let $\sigma(t),\beta(t)$ be the equivalence classes of
$\sigma^{r}(t),\beta^{r}(t)$ mod $2\pi$, i.e.,

$\sigma(t):=\\{\sigma^{r}(t)+2k\pi:k\in\mathbb{Z}\\},\quad\
\beta(t):=\\{\beta^{r}(t)+2k\pi:k\in\mathbb{Z}\\}.$ (3.70)

Then $(\sigma(t),\beta(t),I(t))$ is a function of time, taking values in
$\mathcal{S}_{4}^{*}={\mathbb{T}}^{2}\times(0,I^{*})$, solving (3.68) with
initial datum $(\sigma_{0},\beta_{0},I_{0})$.

If $\chi(h(I_{0}))$ is zero, then both $\Omega_{1}(I_{0})$ and
$\Omega_{2}(I_{0})$ are zero by (3.67), and the solution
$(\sigma(t),\beta(t)$, $I(t))$ is constant in time. If, instead,
$\chi(h(I_{0}))$ is nonzero, then $\Omega_{1}(I_{0})$ is also nonzero by
(3.67) and (3.57), and the function $\sigma(t)$ is periodic in time, with
frequency $\Omega_{1}(I_{0})$ and period
$T_{1}(I_{0}):=2\pi/\Omega_{1}(I_{0})$. By (3.67), and recalling that the
determinant in (3.54) is $=1$, the period $T_{1}(I_{0})$ is equal to the
period $T_{c}$ defined in (3.44) with $c=h(I_{0})$. If $\Omega_{2}(I_{0})$ is
nonzero, then the function $\beta(t)$ is also periodic in time, with frequency
$\Omega_{2}(I_{0})$ and period $T_{2}(I_{0}):=2\pi/\Omega_{2}(I_{0})$.

We observe that the factor $\chi(h(I))$ cancels out in the frequency ratio

$\frac{\Omega_{2}(I)}{\Omega_{1}(I)}=\frac{Q_{0}(I)}{h^{\prime}(I)},$ (3.71)

and $Q_{0}(I)/h^{\prime}(I)$ is well-defined even if $\chi(h(I))$ is zero.
Hence $Q_{0}(I)/h^{\prime}(I)$ is well-defined for all $I\in(0,I^{*})$. By
(3.65) and (3.60),

$\frac{Q_{0}(I)}{h^{\prime}(I)}=\frac{\sqrt{H(h(I))}}{h^{\prime}(I)2\pi}\int_{0}^{2\pi}\frac{1}{\left(1+\sqrt{2\gamma_{h(I)}(g_{h(I)}(\sigma))}\sin(g_{h(I)}(\sigma))\right)^{2}}\,d\sigma.$

We make the change of variable $g_{h(I)}(\sigma)=\vartheta$ in the integral,
that is, $\sigma=f_{c}(\vartheta)$, where $c=h(I)$ and $f_{c}$ is defined in
(3.48). Then $d\sigma=f_{c}^{\prime}(\vartheta)d\vartheta$, and, by (3.48) and
(3.39),
$f_{c}^{\prime}(\vartheta)=\partial_{c}\gamma_{c}(\vartheta)2\pi/F_{c}(2\pi)$.
Hence

$\int_{0}^{2\pi}\frac{1}{\left(1+\sqrt{2\gamma_{h(I)}(g_{h(I)}(\sigma))}\sin(g_{h(I)}(\sigma))\right)^{2}}\,d\sigma=\frac{2\pi}{F_{c}(2\pi)}\int_{0}^{2\pi}\frac{\partial_{c}\gamma_{c}(\vartheta)}{[1+\sqrt{2\gamma_{c}(\vartheta)}\sin\vartheta]^{2}}\,d\vartheta$

where $c=h(I)$. Moreover $h^{\prime}(I)F_{c}(2\pi)=2\pi$ because
$h^{\prime}(I)F_{c}(2\pi)/2\pi$ is the determinant in (3.54), which is $1$.
Therefore

$\displaystyle\frac{Q_{0}(I)}{h^{\prime}(I)}$ $\displaystyle=A(h(I)),\qquad\
A(c):=\sqrt{H(c)}\,J(c),$ $\displaystyle J(c)$
$\displaystyle:=\frac{1}{2\pi}\int_{0}^{2\pi}\frac{\partial_{c}\gamma_{c}(\vartheta)}{[1+\sqrt{2\gamma_{c}(\vartheta)}\sin\vartheta]^{2}}\,d\vartheta.$
(3.72)

###  3.10 Motion of the fluid particles

We consider the composition

$\Phi:=\Phi_{1}\circ\Phi_{2}\circ\Phi_{3}\circ\Phi_{4}\circ\Phi_{5}$ (3.73)

of the transformations defined in (3.3), (3.10), (3.21), (3.52), (3.63). The
map $\Phi$ is defined on
$\mathcal{S}_{4}^{*}={\mathbb{T}}^{2}\times(0,I^{*})$, its image
$\Phi(\mathcal{S}^{*}_{4})$ is the set $\mathcal{S}^{*}$ in (1.10), and $\Phi$
is a diffeomorphism of $\mathcal{S}_{5}^{*}$ onto $\mathcal{S}^{*}$. The image
$(x,y,z)=\Phi(\sigma,\beta,I)\in\mathcal{S}^{*}$ of a point
$(\sigma,\beta,I)\in\mathcal{S}_{4}^{*}$ is

$x=\rho(\sigma,I)\cos(\beta+\eta(\sigma,I)),\quad\
y=\rho(\sigma,I)\sin(\beta+\eta(\sigma,I)),\quad\ z=\zeta(\sigma,I),$

with

$\displaystyle\rho(\sigma,I)$
$\displaystyle:=1+\sqrt{2\gamma_{c}(g_{c}(\sigma))}\sin(g_{c}(\sigma)),\quad\
\zeta(\sigma,I):=\frac{\sqrt{2\gamma_{c}(g_{c}(\sigma))}\cos(g_{c}(\sigma))}{1+\sqrt{2\gamma_{c}(g_{c}(\sigma))}\sin(g_{c}(\sigma))},$
(3.74)

where $c=h(I)$. The analytic regularity of the map $\Phi$ and its expansion
around $I=0$ are studied in Section 4.8.

By construction, a function $\tilde{v}(t)=\Phi(v(t))$ is the solution of the
Cauchy problem (1.11) with initial datum
$\tilde{v}_{0}=\Phi(v_{0})\in\mathcal{S}^{*}$ if and only if the function
$v(t)$ is the solution of (3.64), i.e., (3.68), with initial datum
$v_{0}\in\mathcal{S}_{4}^{*}$. As a consequence, recalling (3.69), (3.70), the
solution $\tilde{v}(t)$ of the Cauchy problem (1.11) with initial datum
$\tilde{v}_{0}=(x_{0},y_{0},z_{0})=\Phi(\sigma_{0},\beta_{0},I_{0})$ is the
function

$\tilde{v}(t)=(x(t),y(t),z(t))=\Phi(\sigma(t),\beta(t),I(t))=\Phi(\sigma_{0}+\Omega_{1}(I_{0})t,\beta_{0}+\Omega_{2}(I_{0})t,I_{0}).$
(3.75)

The function in (3.75) has the form $\tilde{v}(t)=w(\Omega_{1}t,\Omega_{2}t)$
where $w:{\mathbb{T}}^{2}\to\mathcal{S}^{*}$ is the function
$w(\vartheta_{1},\vartheta_{2})=\Phi(\sigma_{0}+\vartheta_{1},\beta_{0}+\vartheta_{2},I_{0})$
and $\Omega_{i}=\Omega_{i}(I_{0})$, $i=1,2$. Hence $\tilde{v}(t)$ is quasi-
periodic with frequency vector $(\Omega_{1},\Omega_{2})$ if $\Omega_{1}\neq
0$, the ratio $\Omega_{2}/\Omega_{1}$ is irrational, and the number of
frequencies cannot be reduced, i.e., if $\tilde{v}(t)$ is not a periodic
function.

Now suppose that $\Omega_{1}$ is nonzero, that $\Omega_{2}/\Omega_{1}$ is
irrational, and that $\tilde{v}(t)$ in (3.75) is periodic with a certain
period $T>0$. Then, by (3.75),

$\begin{pmatrix}\sigma_{0}+\Omega_{1}(t+T)\\\ \beta_{0}+\Omega_{2}(t+T)\\\
I_{0}\end{pmatrix}=\Phi^{-1}(\tilde{v}(t+T))=\Phi^{-1}(\tilde{v}(t))=\begin{pmatrix}\sigma_{0}+\Omega_{1}t\\\
\beta_{0}+\Omega_{2}t\\\ I_{0}\end{pmatrix}\quad\ \forall t\in{\mathbb{R}}.$

Hence $\sigma_{0}+\Omega_{1}(t+T)$ and $\sigma_{0}+\Omega_{1}t$ are the same
element of ${\mathbb{T}}$. This means that $\Omega_{1}T=2\pi n$ for some
$n\in\mathbb{Z}$. Similarly, $\beta_{0}+\Omega_{2}(t+T)=\beta_{0}+\Omega_{2}t$
in ${\mathbb{T}}$, and $\Omega_{2}T=2\pi m$ for some $m\in\mathbb{Z}$. Since
$\Omega_{1}$ is nonzero, $n$ is also nonzero, and $\Omega_{2}/\Omega_{1}=m/n$
is rational, a contradiction. This proves that, for $\Omega_{1}$ nonzero and
$\Omega_{2}/\Omega_{1}$ irrational, the function $\tilde{v}(t)$ in (3.75) is
not periodic, and therefore it is quasi-periodic with frequency vector
$(\Omega_{1},\Omega_{2})$.

###  3.11 The pressure in terms of the action

The pressure and the action are related in the following way. By (1.4), (3.2)
and (3.3), one has
$P(\Phi_{1}(\rho,\varphi,z))=p(\rho,z)=\frac{1}{4}\alpha(\rho,z)$. By (3.10),
(3.13),

$P(\Phi_{1}(\Phi_{2}(\rho,\varphi,z)))=P(\Phi_{1}(\rho,\varphi,z\rho^{-1}))=\frac{1}{4}\alpha(\rho,z\rho^{-1})=\frac{1}{4}\alpha_{2}(\rho,z).$

By (3.21) and (3.23),

$P(\Phi_{1}(\Phi_{2}(\Phi_{3}(\vartheta,\varphi,\xi))))=\frac{1}{4}\alpha_{2}\big{(}1+\sqrt{2\xi}\sin\vartheta,\sqrt{2\xi}\cos\vartheta\big{)}=\frac{1}{4}\alpha_{3}(\vartheta,\xi).$

By (3.31) and (3.52),

$\displaystyle P(\Phi_{1}(\Phi_{2}(\Phi_{3}(\Phi_{4}(\sigma,\varphi,I)))))$
$\displaystyle=\frac{1}{4}\alpha_{3}(g_{c}(\sigma),\gamma_{c}(g_{c}(\sigma)))|_{c=h(I)}=\frac{1}{4}c|_{c=h(I)}=\frac{1}{4}h(I).$

By (3.63) and (3.73),

$P(\Phi(\sigma,\beta,I))=\frac{1}{4}h(I).$ (3.76)

##  4 Taylor expansions and transversality

In this section we prove that the frequency $\Omega_{1}(I)$ and the ratio
$\Omega_{2}(I)/\Omega_{1}(I)$, see (3.67) and (3.71), admit a Taylor expansion
around $I=0$ (which, in principle, is not obvious because of the square roots
in the construction), and we calculate the first nonzero coefficient in their
expansion after the constant term. This forces us to expand the function
$\alpha$ to degree 6. As a consequence, we obtain that $\Omega_{1}(I)$ and
$\Omega_{2}(I)/\Omega_{1}(I)$ really change as $I$ changes, and they vary in a
smooth, strictly monotonic way, passing across every value of an interval
exactly one time. This can be geometrically viewed as a transversality
property. We also calculate the expansion of $\Phi(\sigma,\beta,I)$ around
$I=0$.

###  4.1 Expansion of $\alpha$

The function $\alpha$ is constructed in [7] by solving the pde system (4.4),
where the functions $F$ and $G$ are expressed in terms of a function $\psi$,
see (4.2) and (4.3). The function $\psi$ is defined in [7] as the solution of
a degenerate ode problem. In Section 2.1 of [7] one finds the Taylor expansion
of $\psi$ of order 5 around zero; here we only use its expansion of order 3,
which is

$\psi(s)=1-\frac{3}{4}s+\frac{9}{128}s^{2}-\frac{21}{1024}s^{3}+O(s^{4})\quad\text{as
}s\to 0.$ (4.1)

In Section 2.1 of [7] the functions $H,F,G$ are defined in terms of $\psi$ as

$\displaystyle H(s)$
$\displaystyle=6s\Big{(}\frac{1}{\psi^{\prime}(s)}+2\psi(s)\Big{)},\qquad
F(x,s)=-2x\psi(s)+2x^{3},$ (4.2) $\displaystyle G(x,s)$
$\displaystyle=12x^{2}s-F^{2}(x,s)-H(s).$ (4.3)

In Lemma 3 of [7] the analytic function $\alpha(x,y)$ is defined as the unique
solution of the system

$\partial_{x}\alpha(x,y)=F(x,\alpha(x,y)),\quad\
\big{(}\partial_{y}\alpha(x,y)\big{)}^{2}=G(x,\alpha(x,y))$ (4.4)

in a neighborhood of $(x,y)=(1,0)$ such that $\alpha(1,0)=0$, with
$\partial_{y}\alpha$ not identically zero. In Remark 2 of [7] it is observed
that $\alpha$ is even in $y$, i.e., $\alpha(x,y)=\alpha(x,-y)$. The
coefficients of the monomials of degree 2 and 3 in the Taylor series

$\displaystyle\alpha(x,y)$
$\displaystyle=2(x-1)^{2}+2y^{2}+3(x-1)^{3}+3(x-1)y^{2}+\sum_{\begin{subarray}{c}k,j\geq
0\\\ k+2j\geq 4\end{subarray}}\alpha_{k,2j}(x-1)^{k}y^{2j}$ (4.5)

are given in Remark 4 of [7]; here we want to calculate the coefficients of
the monomials of degree $4,5$ and $6$. Monomials with odd exponent $2j+1$ are
not present in (4.5) because $\alpha$ is even in $y$. One has

$\displaystyle\partial_{x}\alpha(x,y)$
$\displaystyle=4(x-1)+9(x-1)^{2}+3y^{2}+4\alpha_{40}(x-1)^{3}+2\alpha_{22}(x-1)y^{2}$
$\displaystyle\quad\
+5\alpha_{50}(x-1)^{4}+3\alpha_{32}(x-1)^{2}y^{2}+\alpha_{14}y^{4}+6\alpha_{60}(x-1)^{5}$
$\displaystyle\quad\
+4\alpha_{42}(x-1)^{3}y^{2}+2\alpha_{24}(x-1)y^{4}+O_{6},$ (4.6)

where $O_{n}$ denotes terms with homogeneity $\geq n$ in $(x-1,y)$. Since
$\alpha(x,y)=O_{2}$, by (4.1) and (4.5) one has

$\displaystyle\psi(\alpha(x,y))$
$\displaystyle=1-\frac{3}{4}\alpha(x,y)+\frac{9}{128}\alpha^{2}(x,y)+O_{6}$
$\displaystyle=1-\frac{3}{2}(x-1)^{2}-\frac{3}{2}y^{2}-\frac{9}{4}(x-1)^{3}-\frac{9}{4}(x-1)y^{2}+\Big{(}\frac{9}{32}-\frac{3}{4}\alpha_{40}\Big{)}(x-1)^{4}$
$\displaystyle\quad\
+\Big{(}\frac{9}{16}-\frac{3}{4}\alpha_{22}\Big{)}(x-1)^{2}y^{2}+\Big{(}\frac{9}{32}-\frac{3}{4}\alpha_{04}\Big{)}y^{4}+\Big{(}\frac{27}{32}-\frac{3}{4}\alpha_{50}\Big{)}(x-1)^{5}$
$\displaystyle\quad\
+\Big{(}\frac{27}{16}-\frac{3}{4}\alpha_{32}\Big{)}(x-1)^{3}y^{2}+\Big{(}\frac{27}{32}-\frac{3}{4}\alpha_{14}\Big{)}(x-1)y^{4}+O_{6}.$

Expanding $x$ and $x^{3}$ around $x=1$, and recalling the definition (4.2) of
$F$, we calculate

$\displaystyle F(x,\alpha(x,y))$ $\displaystyle=-2x\psi(\alpha(x,y))+2x^{3}$
$\displaystyle=-2\psi(\alpha(x,y))-2(x-1)\psi(\alpha(x,y))+2+6(x-1)+6(x-1)^{2}+2(x-1)^{3}$
$\displaystyle=4(x-1)+9(x-1)^{2}+3y^{2}+\frac{19}{2}(x-1)^{3}+\frac{15}{2}(x-1)y^{2}$
$\displaystyle\quad\
+\Big{(}\frac{63}{16}+\frac{3}{2}\alpha_{40}\Big{)}(x-1)^{4}+\Big{(}\frac{27}{8}+\frac{3}{2}\alpha_{22}\Big{)}(x-1)^{2}y^{2}+\Big{(}-\frac{9}{16}+\frac{3}{2}\alpha_{04}\Big{)}y^{4}$
$\displaystyle\quad\
+\Big{(}-\frac{9}{4}+\frac{3}{2}\alpha_{50}+\frac{3}{2}\alpha_{40}\Big{)}(x-1)^{5}+\Big{(}\frac{9}{4}+\frac{3}{2}\alpha_{32}+\frac{3}{2}\alpha_{22}\Big{)}(x-1)^{3}y^{2}$
$\displaystyle\quad\
+\Big{(}\frac{9}{8}+\frac{3}{2}\alpha_{14}+\frac{3}{2}\alpha_{04}\Big{)}(x-1)y^{4}+O_{6}.$
(4.7)

From (4.6), (4.7) and the identity of each monomial in the differential
equation $\partial_{x}\alpha=F(x,\alpha)$ we get

$\displaystyle\alpha_{40}$ $\displaystyle=\frac{19}{8},\quad\
\alpha_{22}=\frac{15}{4},\quad\ \alpha_{50}=\frac{3}{2},\quad\
\alpha_{32}=3,\quad\ \alpha_{14}=-\frac{9}{16}+\frac{3}{4}\alpha_{04},$
$\displaystyle\alpha_{60}$ $\displaystyle=\frac{19}{32},\quad\
\alpha_{42}=\frac{99}{32},\quad\
\alpha_{24}=\frac{9}{16}+\frac{3}{4}(\alpha_{14}+\alpha_{04}).$

To find the value of $\alpha_{04}$ and $\alpha_{06}$, we consider
$\partial_{y}\alpha(x,y)$ and $G(x,\alpha(x,y))$ at $x=1$ and we expand them
around $y=0$. By (4.5), since $\alpha$ is even in $y$, one has

$\alpha(1,y)=2y^{2}+\alpha_{04}y^{4}+\alpha_{06}y^{6}+O(y^{8}),$ (4.8)

whence

$\displaystyle\partial_{y}\alpha(1,y)$
$\displaystyle=4y+4\alpha_{04}y^{3}+6\alpha_{06}y^{5}+O(y^{7}),$
$\displaystyle(\partial_{y}\alpha(1,y))^{2}$
$\displaystyle=16y^{2}+32\alpha_{04}y^{4}+(48\alpha_{06}+16\alpha_{04}^{2})y^{6}+O(y^{8}).$
(4.9)

By (4.7),

$\displaystyle F(1,\alpha(1,y))$
$\displaystyle=3y^{2}+\Big{(}-\frac{9}{16}+\frac{3}{2}\alpha_{04}\Big{)}y^{4}+O(y^{6}),$
$\displaystyle F^{2}(1,\alpha(1,y))$
$\displaystyle=9y^{4}+\Big{(}-\frac{27}{8}+9\alpha_{04}\Big{)}y^{6}+O(y^{8}).$

By (4.1) and (4.2),

$H(s)=4s-\frac{21}{2}s^{2}+\frac{39}{32}s^{3}+O(s^{4})$ (4.10)

(it is to obtain (4.10) that we use the coefficient of $s^{3}$ in (4.1)).
Therefore, by (4.8),

$H(\alpha(1,y))=8y^{2}+(4\alpha_{04}-42)y^{4}+\Big{(}4\alpha_{06}-42\alpha_{04}+\frac{39}{4}\Big{)}y^{6}+O(y^{8}).$

Hence

$\displaystyle G(1,\alpha(1,y))$
$\displaystyle=12\alpha(1,y)-F^{2}(1,\alpha(1,y))-H(\alpha(1,y))$
$\displaystyle=16y^{2}+(8\alpha_{04}+33)y^{4}+\Big{(}8\alpha_{06}+33\alpha_{04}+\frac{105}{8}\Big{)}y^{6}+O(y^{8}).$
(4.11)

By (4.9), (4.11) and the identity of each monomial in the differential
equation $(\partial_{y}\alpha)^{2}=G(x,\alpha)$ at $x=1$ we get
$\alpha_{04}=11/8$ and $\alpha_{06}=113/160$. Hence $\alpha_{14}=15/32$,
$\alpha_{24}=249/128$, and

$\displaystyle\alpha(x,y)$
$\displaystyle=2(x-1)^{2}+2y^{2}+3(x-1)^{3}+3(x-1)y^{2}+\frac{19}{8}(x-1)^{4}+\frac{15}{4}(x-1)^{2}y^{2}$
$\displaystyle\quad\
+\frac{11}{8}y^{4}+\frac{3}{2}(x-1)^{5}+3(x-1)^{3}y^{2}+\frac{15}{32}(x-1)y^{4}+\frac{19}{32}(x-1)^{6}$
$\displaystyle\quad\
+\frac{99}{32}(x-1)^{4}y^{2}+\frac{249}{128}(x-1)^{2}y^{4}+\frac{113}{160}y^{6}+O_{7}.$
(4.12)

###  4.2 Expansion of $\alpha_{2}$

The function $\alpha_{2}(\rho,z)=\alpha(\rho,z/\rho)$ defined in (3.13) is
even in $z$, it is analytic around $(\rho,z)=(1,0)$, and its Taylor series

$\alpha_{2}(\rho,z)=\sum_{\begin{subarray}{c}k,j\geq 0\\\ k+2j\geq
2\end{subarray}}(\alpha_{2})_{k,2j}(\rho-1)^{k}y^{2j}$ (4.13)

matches with the expansion in (4.12) in which $x=\rho$ and $y=z/\rho$. We
expand

$\displaystyle\rho^{-2}$
$\displaystyle=1-2(\rho-1)+3(\rho-1)^{2}-4(\rho-1)^{3}+5(\rho-1)^{4}+O((\rho-1)^{5}),$
$\displaystyle\rho^{-4}$
$\displaystyle=1-4(\rho-1)+10(\rho-1)^{2}+O((\rho-1)^{3}),$

and we obtain

$\displaystyle\alpha_{2}(\rho,z)$
$\displaystyle=2(\rho-1)^{2}+2z^{2}+3(\rho-1)^{3}-(\rho-1)z^{2}+\frac{19}{8}(\rho-1)^{4}+\frac{15}{4}(\rho-1)^{2}y^{2}$
$\displaystyle\quad\
+\frac{11}{8}z^{4}+\frac{3}{2}(\rho-1)^{5}-\frac{7}{2}(\rho-1)^{3}z^{2}-\frac{161}{32}(\rho-1)z^{4}+\frac{19}{32}(\rho-1)^{6}$
$\displaystyle\quad\
+\frac{203}{32}(\rho-1)^{4}z^{2}+\frac{1529}{128}(\rho-1)^{2}z^{4}+\frac{113}{160}z^{6}+O_{7}.$
(4.14)

###  4.3 Expansion of $\gamma_{c}(\vartheta)$

Recalling the definition (3.18) of $\mathcal{B}_{2}$ and the first inclusion
in (3.19), the function $\alpha_{2}(\rho,z)$ in (3.13) is defined and analytic
in the disc $(\rho-1)^{2}+z^{2}$ $<\delta_{2}^{2}$. We define

$\phi:{\mathbb{T}}\times(-\delta_{2},\delta_{2})\to{\mathbb{R}}^{2},\quad\
\phi(\vartheta,r):=(1+r\sin\vartheta,\,r\cos\vartheta),\quad\
\widetilde{\alpha}_{3}(\vartheta,r):=\alpha_{2}(\phi(\vartheta,r)).$ (4.15)

The function $\widetilde{\alpha}_{3}$ is well-posed and analytic in
$(\vartheta,r)\in{\mathbb{T}}\times(-\delta_{2},\delta_{2})$, and, by (4.13),
it is the power series

$\widetilde{\alpha}_{3}(\vartheta,r)=\sum_{n=2}^{\infty}P_{n}(\vartheta)r^{n},\quad\
P_{n}(\vartheta)=\sum_{\begin{subarray}{c}k,j\geq 0\\\
k+2j=n\end{subarray}}(\alpha_{2})_{k,2j}(\sin\vartheta)^{k}(\cos\vartheta)^{2j}.$
(4.16)

All $P_{n}(\vartheta)$ are trigonometric polynomials, and
$P_{n}(-\vartheta)=(-1)^{n}P_{n}(\vartheta)$, i.e., $P_{n}$ is even for $n$
even and $P_{n}$ is odd for $n$ odd, because $(-1)^{k}=(-1)^{n}$ for $k+2j=n$.
Hence

$\widetilde{\alpha}_{3}(-\vartheta,-r)=\widetilde{\alpha}_{3}(\vartheta,r)\quad\
\forall(\vartheta,r)\in{\mathbb{T}}\times(-\delta_{2},\delta_{2}),$ (4.17)

namely $\widetilde{\alpha}_{3}$ is an even function of the pair
$(\vartheta,r)$. In fact, (4.17) is the symmetry property
$\alpha_{2}(\rho,-z)=\alpha_{2}(\rho,z)$ expressed in terms of the function
$\widetilde{\alpha}_{3}(\vartheta,r)$. By (4.14),

$\displaystyle P_{2}(\vartheta)$ $\displaystyle=2,$ (4.18) $\displaystyle
P_{3}(\vartheta)$
$\displaystyle=3\sin^{3}\vartheta-\sin\vartheta\cos^{2}\vartheta,$
$\displaystyle P_{4}(\vartheta)$
$\displaystyle=\frac{19}{8}\sin^{4}\vartheta+\frac{15}{4}\sin^{2}\vartheta\cos^{2}\vartheta+\frac{11}{8}\cos^{4}\vartheta,$
$\displaystyle P_{5}(\vartheta)$
$\displaystyle=\frac{3}{2}\sin^{5}\vartheta-\frac{7}{2}\sin^{3}\vartheta\cos^{2}\vartheta-\frac{161}{32}\sin\vartheta\cos^{4}\vartheta,$
$\displaystyle P_{6}(\vartheta)$
$\displaystyle=\frac{19}{32}\sin^{6}\vartheta+\frac{203}{32}\sin^{4}\vartheta\cos^{2}\vartheta+\frac{1529}{128}\sin^{2}\vartheta\cos^{4}\vartheta+\frac{113}{160}\cos^{6}\vartheta.$

Hence

$\displaystyle P_{3}(\vartheta)$
$\displaystyle=2\sin(\vartheta)-\sin(3\vartheta),$ (4.19) $\displaystyle
P_{4}(\vartheta)$ $\displaystyle=\frac{15}{8}-\frac{1}{2}\cos(2\vartheta),$
(4.20) $\displaystyle P_{5}(\vartheta)$
$\displaystyle=-\frac{33}{256}\sin(\vartheta)-\frac{835}{512}\sin(3\vartheta)+(P_{5})_{5}\sin(5\vartheta),$
(4.21) $\displaystyle P_{6}(\vartheta)$
$\displaystyle=\frac{3173}{2048}+(P_{6})_{2}\cos(2\vartheta)+(P_{6})_{4}\cos(4\vartheta)+(P_{6})_{6}\cos(6\vartheta).$
(4.22)

The Fourier coefficients $(P_{5})_{5}$ in (4.21) and $(P_{6})_{k}$, $k=2,4,6$,
in (4.22) are not involved in the calculations we are going to make, and
therefore we avoid to calculate their numerical value.

When $\widetilde{\alpha}_{3}(\vartheta,r)$ is evaluated at $r=\sqrt{2\xi}$, we
obtain the function $\alpha_{3}(\vartheta,\xi)$ defined in (3.23), i.e.,

$\alpha_{3}(\vartheta,\xi)=\widetilde{\alpha}_{3}(\vartheta,\sqrt{2\xi}).$
(4.23)

The function $\gamma_{c}(\vartheta)$ is defined in (3.31) as the unique
solution $\xi$ of the equation $\alpha_{3}(\vartheta,\xi)=c$. Because of the
square root in the construction, $\gamma_{c}(\vartheta)$, as a function of
$c$, is not analytic around $c=0$, i.e., it is not a power series of the form
$\sum Q_{n}(\vartheta)c^{n}$ for some analytic functions $Q_{n}(\vartheta)$.
However, $\gamma_{c}(\vartheta)$ is a power series of the form $\sum
Q_{n}(\vartheta)c^{n/2}$, namely there exists a function
$\widetilde{\gamma}(\vartheta,\mu)$, analytic around $\mu=0$, such that
$\gamma_{c}(\vartheta)$ is $\widetilde{\gamma}(\vartheta,\mu)$ evaluated at
$\mu=\sqrt{c}$. To prove it, we use the implicit function theorem for analytic
functions, taking into account the degeneracy of the problem.

We define

$\mathcal{F}(\vartheta,\mu,w):=\mu^{-2}\widetilde{\alpha}_{3}(\vartheta,\mu
w)-1\quad\ \text{if }\mu\neq 0;\qquad\ \mathcal{F}(\vartheta,0,w):=2w^{2}-1.$
(4.24)

The function $\mathcal{F}$ in (4.24) is well-defined and analytic in
${\mathbb{T}}\times(-\mu_{0},\mu_{0})\times(-w_{0},w_{0})$, for some
$\mu_{0},w_{0}>0$ small enough, and, by (4.16),

$\displaystyle\mathcal{F}(\vartheta,\mu,w)$
$\displaystyle=\sum_{n=2}^{\infty}P_{n}(\vartheta)w^{n}\mu^{n-2}-1=2w^{2}-1+\sum_{n=3}^{\infty}P_{n}(\vartheta)w^{n}\mu^{n-2}.$

Moreover, by (4.17) and (4.24),

$\mathcal{F}(-\vartheta,-\mu,w)=\mathcal{F}(\vartheta,\mu,w)\quad\
\forall(\vartheta,\mu,w)\in{\mathbb{T}}\times(-\mu_{0},\mu_{0})\times(-w_{0},w_{0}).$
(4.25)

For every $\vartheta\in{\mathbb{T}}$ one has

$\mathcal{F}(\vartheta,0,2^{-\frac{1}{2}})=0,\quad\
\partial_{w}\mathcal{F}(\vartheta,0,2^{-\frac{1}{2}})=4\cdot
2^{-\frac{1}{2}}\neq 0.$

Hence there exist two constants $\mu_{1},w_{1}$, with $0<\mu_{1}\leq\mu_{0}$,
$0<w_{1}\leq w_{0}$, and a function $w(\vartheta,\mu)$, defined and analytic
in ${\mathbb{T}}\times(-\mu_{1},\mu_{1})$, taking values in $(-w_{1},w_{1})$,
such that

$w(\vartheta,0)=2^{-\frac{1}{2}}\quad\
\forall\vartheta\in{\mathbb{T}},\qquad\mathcal{F}(\vartheta,\mu,w(\vartheta,\mu))=0\quad\
\forall(\vartheta,\mu)\in{\mathbb{T}}\times(-\mu_{1},\mu_{1}),$ (4.26)

and such that if a point
$(\vartheta,\mu,a)\in{\mathbb{T}}\times(-\mu_{1},\mu_{1})\times(-w_{1},w_{1})$
is a zero of $\mathcal{F}$, then $a=w(\vartheta,\mu)$.

By (4.26) and (4.25), for all $(\vartheta,\mu)$ one has

$0=\mathcal{F}(-\vartheta,-\mu,w(-\vartheta,-\mu))=\mathcal{F}(\vartheta,\mu,w(-\vartheta,-\mu)).$

Hence the point
$(\vartheta,\mu,w(-\vartheta,-\mu))\in{\mathbb{T}}\times(-\mu_{1},\mu_{1})\times(-w_{1},w_{1})$
is a zero of $\mathcal{F}$, and therefore it belongs to the graph of the
implicit function, i.e.,

$w(-\vartheta,-\mu)=w(\vartheta,\mu)\quad\
\forall(\vartheta,\mu)\in{\mathbb{T}}\times(-\mu_{1},\mu_{1}).$ (4.27)

From the second identity in (4.26) and formula (4.16) it follows that
$w(\vartheta,r)$ is the power series

$w(\vartheta,\mu)=\sum_{n=0}^{\infty}W_{n}(\vartheta)\mu^{n},$ (4.28)

where the functions $W_{n}(\vartheta)$ are determined by the identity
$\sum_{n=2}^{\infty}P_{n}(\vartheta)w^{n}(\vartheta,\mu)\mu^{n-2}=1$, i.e.,
$W_{0}(\vartheta)=2^{-\frac{1}{2}}$ and $W_{n}(\vartheta)$ are trigonometric
polynomials recursively determined by the system

$\sum_{\begin{subarray}{c}n\geq 2,\ j\geq 0,\\\ n-2+j=m\end{subarray}}\
\sum_{\begin{subarray}{c}k_{1},\ldots,k_{n}\geq 0,\\\
k_{1}+\ldots+k_{n}=j\end{subarray}}P_{n}(\vartheta)W_{k_{1}}(\vartheta)W_{k_{2}}(\vartheta)\cdots
W_{k_{n}}(\vartheta)=0\quad\ \forall m\geq 1.$ (4.29)

By (4.27) and (4.28), $W_{n}(-\vartheta)=(-1)^{n}W_{n}(\vartheta)$, i.e.,
$W_{n}$ is even for $n$ even and $W_{n}$ is odd for $n$ odd. We will make use
of equations (4.29) for $m=1,2,3,4$, which are

$\displaystyle 2P_{2}W_{0}W_{1}+P_{3}W_{0}^{3}=0,$ (4.30) $\displaystyle
P_{2}(2W_{0}W_{2}+W_{1}^{2})+3P_{3}W_{0}^{2}W_{1}+P_{4}W_{0}^{4}=0,$ (4.31)
$\displaystyle
P_{2}(2W_{0}W_{3}+2W_{1}W_{2})+P_{3}(3W_{0}^{2}W_{2}+3W_{0}W_{1}^{2})+4P_{4}W_{0}^{3}W_{1}+P_{5}W_{0}^{5}=0,$
(4.32) $\displaystyle
P_{2}(2W_{0}W_{4}+2W_{1}W_{3}+W_{2}^{2})+P_{3}(3W_{0}^{2}W_{3}+6W_{0}W_{1}W_{2}+W_{1}^{3})$
$\displaystyle+P_{4}(4W_{0}^{3}W_{2}+6W_{0}^{2}W_{1}^{2})+5P_{5}W_{0}^{4}W_{1}+P_{6}W_{0}^{6}=0.$
(4.33)

By the definition (4.24) of $\mathcal{F}$, the second identity in (4.26)
implies that

$\widetilde{\alpha}_{3}(\vartheta,\mu w(\vartheta,\mu))=\mu^{2}\quad\
\forall(\vartheta,\mu)\in{\mathbb{T}}\times(-\mu_{1},\mu_{1})$ (4.34)

with $\mu\neq 0$. Identity (4.34) also holds for $\mu=0$ because
$\widetilde{\alpha}_{3}(\vartheta,0)=\alpha_{2}(\phi(\vartheta,0))=\alpha_{2}(1,0)=0$.

By the first identity in (4.26), taking $\mu_{1}$ smaller if necessary, one
has $w(\vartheta,\mu)>0$ for all
$(\vartheta,\mu)\in{\mathbb{T}}\times(-\mu_{1},\mu_{1})$. Given any
$c\in[0,\mu_{1}^{2})$, there exists a unique $\mu\in[0,\mu_{1})$ such that
$c=\mu^{2}$, that is, $\mu=\sqrt{c}$. Also, $\mu w(\vartheta,\mu)\geq 0$, and
there exists a unique $\xi\geq 0$ such that $\mu
w(\vartheta,\mu)=\sqrt{2\xi}$, that is,
$\xi=\frac{1}{2}\mu^{2}w^{2}(\vartheta,\mu)$. As a consequence, by (4.23) and
(4.34),

$\alpha_{3}(\vartheta,\xi)=\widetilde{\alpha}_{3}(\vartheta,\sqrt{2\xi})=\widetilde{\alpha}_{3}(\vartheta,\mu
w(\vartheta,\mu))=\mu^{2}=c.$ (4.35)

Hence $\xi=\gamma_{c}(\vartheta)$, and therefore
$\gamma_{c}(\vartheta)=\frac{1}{2}\mu^{2}w^{2}(\vartheta,\mu)$ where
$\mu=\sqrt{c}$. In other words, we have proved that

$\gamma_{c}(\vartheta)=\widetilde{\gamma}(\vartheta,\sqrt{c})\quad\
\forall(\vartheta,c)\in{\mathbb{T}}\times[0,\mu_{1}^{2}),$ (4.36)

where $\widetilde{\gamma}$ is the analytic function

$\widetilde{\gamma}(\vartheta,\mu):=\frac{1}{2}\mu^{2}w^{2}(\vartheta,\mu).$
(4.37)

By (4.27),

$\widetilde{\gamma}(-\vartheta,-\mu)=\widetilde{\gamma}(\vartheta,\mu)\quad\
\forall(\vartheta,\mu)\in{\mathbb{T}}\times(-\mu_{1},\mu_{1}).$ (4.38)

By (4.37) and (4.28),

$\widetilde{\gamma}(\vartheta,\mu)=\sum_{n=2}^{\infty}Q_{n}(\vartheta)\mu^{n},\quad\
Q_{n}(\vartheta):=\frac{1}{2}\sum_{\begin{subarray}{c}k,j\geq 0\\\
k+j+2=n\end{subarray}}W_{k}(\vartheta)W_{j}(\vartheta).$ (4.39)

By (4.38) and (4.39), one has $Q_{n}(-\vartheta)=(-1)^{n}Q_{n}(\vartheta)$.

###  4.4 Expansion of the average of $\gamma_{c}(\vartheta)$ and
$\partial_{c}\gamma_{c}(\vartheta)$

We study the average of $\widetilde{\gamma}(\vartheta,\mu)$,
$\gamma_{c}(\vartheta)$ and $\partial_{c}\gamma_{c}(\vartheta)$ over
$\vartheta\in[0,2\pi]$. To shorten the notation, given any $2\pi$-periodic
function $f(\vartheta)$, we denote $\langle f\rangle$ its average over the
period, i.e.,

$\langle f\rangle:=\frac{1}{2\pi}\int_{0}^{2\pi}f(\vartheta)\,d\vartheta.$

For $n$ odd, the trigonometric polynomial $Q_{n}(\vartheta)$ in (4.39) is
$2\pi$-periodic and odd, and therefore $\langle Q_{n}\rangle=0$. As a
consequence, by (4.39),

$\frac{1}{2\pi}\int_{0}^{2\pi}\widetilde{\gamma}(\vartheta,\mu)\,d\vartheta=\sum_{k=1}^{\infty}\langle
Q_{2k}\rangle\mu^{2k}.$ (4.40)

By (3.55), (4.36) and (4.40), one has

$h_{1}(c)=\frac{1}{2\pi}\int_{0}^{2\pi}\gamma_{c}(\vartheta)\,d\vartheta=\frac{1}{2\pi}\int_{0}^{2\pi}\widetilde{\gamma}(\vartheta,\sqrt{c})\,d\vartheta=\sum_{k=1}^{\infty}\langle
Q_{2k}\rangle c^{k}.$ (4.41)

Hence $h_{1}(c)$ is analytic around $c=0$, in the sense that $h_{1}(c)$, which
is defined for $c\in[0,\mu_{1}^{2})$, coincides in $[0,\mu_{1}^{2})$ with the
power series in (4.41), which is a function defined for
$c\in(-\mu_{1}^{2},\mu_{1}^{2})$ and analytic in that interval. Note that the
average of $\gamma_{c}(\vartheta)$ is analytic around $c=0$ even if the
function $\gamma_{c}(\vartheta)$ itself is not analytic in $c$ around $c=0$.

Now we calculate the averages $\langle Q_{n}\rangle$ for $n=2,4,6$. By (4.39),

$Q_{2}=\frac{1}{2}W_{0}^{2}=\frac{1}{4},\quad\
Q_{4}=\frac{1}{2}(2W_{0}W_{2}+W_{1}^{2}),\quad\
Q_{6}=\frac{1}{2}(2W_{0}W_{4}+2W_{1}W_{3}+W_{2}^{2}).$ (4.42)

Hence $\langle Q_{2}\rangle=1/4$. Since $P_{2}=2$ and
$W_{0}=2^{-\frac{1}{2}}$, from (4.30) we get

$W_{1}=-\frac{1}{8}P_{3}.$ (4.43)

Using (4.31) to substitute $(2W_{0}W_{2}+W_{1}^{2})$, and using also (4.43),
we calculate

$Q_{4}=-\frac{1}{4}(3P_{3}W_{0}^{2}W_{1}+P_{4}W_{0}^{4})=\frac{1}{64}(3P_{3}^{2}-4P_{4}).$

By (4.19) and (4.20), we obtain

$\langle Q_{4}\rangle=0.$ (4.44)

Using (4.33) to substitute $(2W_{0}W_{4}+2W_{1}W_{3}+W_{2}^{2})$, and also
(4.43), we calculate

$\displaystyle Q_{6}$
$\displaystyle=-\frac{1}{4}\big{[}P_{3}(3W_{0}^{2}W_{3}+6W_{0}W_{1}W_{2}+W_{1}^{3})+P_{4}(4W_{0}^{3}W_{2}+6W_{0}^{2}W_{1}^{2})+5P_{5}W_{0}^{4}W_{1}+P_{6}W_{0}^{6}\big{]}$
$\displaystyle=-\frac{3}{8}P_{3}W_{3}+\frac{3\sqrt{2}}{32}P_{3}^{2}W_{2}+\frac{1}{2048}P_{3}^{4}-\frac{\sqrt{2}}{4}P_{4}W_{2}-\frac{3}{256}P_{3}^{2}P_{4}+\frac{5}{128}P_{3}P_{5}-\frac{1}{32}P_{6}.$
(4.45)

From (4.31) we obtain

$\displaystyle W_{2}$
$\displaystyle=\frac{5\sqrt{2}}{128}P_{3}^{2}-\frac{\sqrt{2}}{16}P_{4}=\frac{\sqrt{2}}{256}\Big{(}-5-32\cos(2\vartheta)+20\cos(4\vartheta)-5\cos(6\vartheta)\Big{)}$
(4.46)

and, from (4.32) and (4.43),

$\displaystyle W_{3}$
$\displaystyle=\frac{P_{3}P_{4}}{16}-\frac{P_{3}W_{2}}{2\sqrt{2}}-\frac{3P_{3}^{3}}{256}-\frac{P_{5}}{16}=\frac{225}{4096}\sin(\vartheta)+\frac{1251}{8192}\sin(3\vartheta)+\sum_{k=5,7,9}(W_{3})_{k}\sin(k\vartheta).\,$
(4.47)

The Fourier coefficients $(W_{3})_{k}$, $k=5,7,9$, are not involved in the
calculation of $\langle Q_{6}\rangle$, and therefore we do not calculate them.
By (4.19), …, (4.22), (4.46), (4.47), we calculate

$\displaystyle\langle P_{3}W_{3}\rangle$
$\displaystyle=-\frac{351}{16384},\qquad$ $\displaystyle\langle
P_{3}^{2}W_{2}\rangle$ $\displaystyle=\frac{291\sqrt{2}}{1024},\qquad$
$\displaystyle\langle P_{3}^{4}\rangle$ $\displaystyle=\frac{131}{8},\qquad$
$\displaystyle\langle P_{4}W_{2}\rangle$
$\displaystyle=-\frac{11\sqrt{2}}{2048},$ $\displaystyle\langle
P_{3}^{2}P_{4}\rangle$ $\displaystyle=\frac{91}{16},\qquad$
$\displaystyle\langle P_{3}P_{5}\rangle$
$\displaystyle=\frac{703}{1024},\qquad$ $\displaystyle\langle P_{6}\rangle$
$\displaystyle=\frac{3173}{2048}.$

Hence, integrating (4.45), we obtain

$\displaystyle\langle Q_{6}\rangle$ $\displaystyle=-\frac{1065}{65536}$ (4.48)

(where $65536=2^{16}$), and, by (4.41),

$h_{1}(c)=\frac{1}{2\pi}\int_{0}^{2\pi}\gamma_{c}(\vartheta)\,d\vartheta=\frac{c}{4}+\langle
Q_{6}\rangle c^{3}+O(c^{4}).$ (4.49)

Since $\langle Q_{6}\rangle$ in (4.48) is nonzero, $h_{1}(c)$ is a nonlinear,
analytic function of $c$. Its derivative is

$h_{1}^{\prime}(c)=\frac{1}{2\pi}\int_{0}^{2\pi}\partial_{c}\gamma_{c}(\vartheta)\,d\vartheta=\frac{1}{4}+3\langle
Q_{6}\rangle c^{2}+O(c^{3}),$ (4.50)

which is a nonconstant, analytic function of $c$.

###  4.5 Expansion of $J(c)$

The average $J(c)$ is defined in (3.72). By (4.36), the partial derivative
$\partial_{c}\gamma_{c}(\vartheta)$ satisfies

$\partial_{c}\gamma_{c}(\vartheta)=\nu(\vartheta,\sqrt{c}),$ (4.51)

where $\nu(\vartheta,\mu)$ is the analytic function

$\nu(\vartheta,\mu):=\frac{\partial_{\mu}\widetilde{\gamma}(\vartheta,\mu)}{2\mu}=\sum_{n=2}^{\infty}Q_{n}(\vartheta)\frac{n}{2}\mu^{n-2},$
(4.52)

with $Q_{n}(\vartheta)$ defined in (4.39). By (4.38), one has
$\nu(-\vartheta,-\mu)=\nu(\vartheta,\mu)$. Moreover

$\nu(\vartheta,\mu)=Q_{2}+\frac{3}{2}Q_{3}(\vartheta)\mu+2Q_{4}(\vartheta)\mu^{2}+O(\mu^{3}),$
(4.53)

and $Q_{2}=1/4$ by (4.42), $Q_{3}=W_{0}W_{1}$ by (4.39), $\langle
Q_{3}\rangle=0$ because $Q_{3}$ is odd, and $\langle Q_{4}\rangle=0$ by
(4.44). Regarding the denominator in the definition of $J(c)$, one has
$\sqrt{2\gamma_{c}(\vartheta)}=\sqrt{2\xi}=\sqrt{c}\,w(\vartheta,\sqrt{c})$ by
construction (see (4.35) and the lines preceding it), and, by (4.28), (4.43),
(4.19),

$\displaystyle\big{(}1+\mu w(\vartheta,\mu)\sin\vartheta\big{)}^{-2}$
$\displaystyle=1-2\mu
w(\vartheta,\mu)\sin\vartheta+3\mu^{2}w^{2}(\vartheta,\mu)\sin^{2}\vartheta+O(\mu^{3})$
$\displaystyle=1-2W_{0}\sin(\vartheta)\mu+\big{(}3W_{0}^{2}\sin^{2}\vartheta-2W_{1}(\vartheta)\sin\vartheta\big{)}\mu^{2}+O(\mu^{3})$
$\displaystyle=1-\sqrt{2}\sin(\vartheta)\mu+\Big{(}1-\frac{9}{8}\cos(2\vartheta)+\frac{1}{8}\cos(4\vartheta)\Big{)}\mu^{2}+O(\mu^{3}).$
(4.54)

Taking a smaller $\mu_{1}$ if necessary, the function

$m(\vartheta,\mu):=\frac{\nu(\vartheta,\mu)}{[1+\mu
w(\vartheta,\mu)\sin\vartheta]^{2}}$ (4.55)

is defined and analytic in ${\mathbb{T}}\times(-\mu_{1},\mu_{1})$, it
satisfies

$m(-\vartheta,-\mu)=m(\vartheta,\mu)\quad\
\forall(\vartheta,\mu)\in{\mathbb{T}}\times(-\mu_{1},\mu_{1}),$ (4.56)

and it has expansion
$m(\vartheta,\mu)=\sum_{n=0}^{\infty}M_{n}(\vartheta)\mu^{n}$ for some
trigonometric polynomials $M_{n}(\vartheta)$. From (4.56) it follows that
$M_{n}(-\vartheta)=(-1)^{n}M_{n}(\vartheta)$. Therefore $\langle
M_{n}\rangle=0$ for $n$ odd, and

$\frac{1}{2\pi}\int_{0}^{2\pi}m(\vartheta,\mu)\,d\vartheta=\sum_{k=0}^{\infty}\langle
M_{2k}\rangle\mu^{2k}.$

By (4.53) and (4.54) one has

$M_{0}=\frac{1}{4},\quad\
M_{2}(\vartheta)=2Q_{4}(\vartheta)-\frac{3}{2}W_{1}(\vartheta)\sin\vartheta+\frac{1}{4}-\frac{9}{32}\cos(2\vartheta)+\frac{1}{32}\cos(4\vartheta).$

By (4.43) and (4.19), the average of $W_{1}(\vartheta)\sin\vartheta$ is
$-1/8$. Hence $\langle M_{2}\rangle=7/16$, and

$\displaystyle J(c)$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}\frac{\partial_{c}\gamma_{c}(\vartheta)}{[1+\sqrt{2\gamma_{c}(\vartheta)}\sin\vartheta]^{2}}\,d\vartheta$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}m(\vartheta,\sqrt{c})\,d\vartheta=\sum_{k=0}^{\infty}\langle
M_{2k}\rangle c^{k}=\frac{1}{4}+\frac{7}{16}c+O(c^{2}).$ (4.57)

Thus, the average $J(c)$ is an analytic function of $c$ around $c=0$ (even if
the integrand function $m(\vartheta,\sqrt{c})$ is not). Moreover, taking
$\mu_{1}$ smaller if necessary, $J(c)$ is strictly increasing in
$[0,\mu_{1}^{2})$.

###  4.6 Expansion of $h(I)$ and $\Omega_{1}(I)$

We have already proved that $h_{1}(c)$ in (3.55) is analytic around $c=0$,
with expansion (4.49). Hence its inverse function $h(I)=h_{1}^{-1}(I)$ is also
analytic around $I=0$, and it satisfies

$h(I)=4I-256\langle Q_{6}\rangle I^{3}+O(I^{4}),\quad\
h^{\prime}(I)=4-768\langle Q_{6}\rangle I^{2}+O(I^{3}).$ (4.58)

The frequency $\Omega_{1}(I)$ is defined in (3.67), and it is the product of
the $C^{\infty}$ cut-off function $\chi(h(I))$ times the analytic function
$h^{\prime}(I)$ in (4.58).

###  4.7 Expansion of the frequency ratio $\Omega_{2}(I)/\Omega_{1}(I)$

By (3.71) and (3.72), the frequency ratio $\Omega_{2}(I)/\Omega_{1}(I)$
coincides with the function

$A(h(I))=\sqrt{H(h(I))}\,J(h(I)).$

By its definition in [7], the function $H(c)$ is analytic around $c=0$. Hence
the composition $H(h(I))$ is analytic around $I=0$, and, by (4.10) and (4.58),

$H(h(I))=16I-168I^{2}+O(I^{3}).$

We write its square root as the product

$\sqrt{H(h(I))}=\sqrt{I}\,B(I),$ (4.59)

where

$B(I):=4\Big{(}\frac{H(h(I))}{16I}\Big{)}^{\frac{1}{2}}=4\Big{(}1-\frac{21}{2}I+O(I^{2})\Big{)}^{\frac{1}{2}}=4-21I+O(I^{2}).$
(4.60)

Since the function $x\mapsto\sqrt{1+x}$ is analytic around $x=0$, the function
$B(I)$ is analytic around $I=0$. The function $J(h(I))$ is also analytic
around $I=0$, and, by (4.57) and (4.58),

$J(h(I))=\frac{1}{4}+\frac{7}{4}I+O(I^{2}).$ (4.61)

Hence

$A(h(I))=\sqrt{I}\,\mathcal{R}(I),\quad\
\mathcal{R}(I):=B(I)J(h(I))=1+\frac{7}{4}I+O(I^{2}),$

and the function $\mathcal{R}(I)$ is analytic around $I=0$. Taking a smaller
$I^{*}$ if necessary, both $\mathcal{R}(I)$ and $A(h(I))$ are strictly
increasing functions of $I\in[0,I^{*})$.

###  4.8 Expansion of $\Phi(\sigma,\beta,I)$

Since $Q_{2}=1/4$, by (4.36), (4.39), (4.51), (4.52) one has

$\gamma_{c}(\vartheta)=\frac{c}{4}+O(c^{\frac{3}{2}}),\quad\
\partial_{c}\gamma_{c}(\vartheta)=\frac{1}{4}+O(c^{\frac{1}{2}}).$

Hence, by (3.39), $F_{c}(\vartheta)=\frac{\vartheta}{4}+O(c^{\frac{1}{2}})$
and therefore, by (3.48), $f_{c}(\vartheta)=\vartheta+O(c^{\frac{1}{2}})$. As
a consequence, $g_{c}$ in (3.49), being the inverse of $f_{c}$, satisfies

$g_{c}(\vartheta)=\vartheta+O(c^{\frac{1}{2}}).$

By (4.58), $h(I)=4I+O(I^{3})$ and $h^{\prime}(I)=4+O(I^{2})$. Hence, at
$c=h(I)$, one has

$\gamma_{c}(\vartheta)=I+O(I^{\frac{3}{2}}),\qquad\
g_{c}(\vartheta)=\vartheta+O(I^{\frac{1}{2}}),$

and therefore the functions $\rho(\sigma,I),\zeta(\sigma,I)$ in (3.74) have
expansion

$\rho(\sigma,I)=\sqrt{2I}\,\sin(\sigma)+O(I),\qquad\
\zeta(\sigma,I)=\sqrt{2I}\,\cos(\sigma)+O(I).$ (4.62)

By (4.59) and (4.60),

$\sqrt{H(h(I))}=4\sqrt{I}+O(I^{\frac{3}{2}}).$ (4.63)

By (4.62) and (4.63), the function $Q(\sigma,I)$ in (3.60) satisfies

$Q(\sigma,I)=4I^{\frac{1}{2}}+O(I).$

Hence $Q_{0},\widetilde{Q},\eta$ in (3.65), (3.66) satisfy

$Q_{0}(I)=4I^{\frac{1}{2}}+O(I),\quad\ \widetilde{Q}(\sigma,I)=O(I),\quad\
\eta(\sigma,I)=O(I).$

The map $\Phi_{1}$ in (3.3) is analytic in $\mathcal{N}_{1}$ because $\rho$
does not vanish in $\mathcal{N}_{1}$; the map $\Phi_{2}$ in (3.10) is also
analytic in $\mathcal{N}_{2}$ because $\rho$ does not vanish in
$\mathcal{N}_{2}$.

The map $\Phi_{3}$ in (3.21) is analytic in $\mathcal{B}_{3}$ because $\xi$ is
positive in $\mathcal{B}_{3}$; however, $\Phi_{3}$ is not analytic in $\xi$
around $\xi=0$. Nonetheless, $\Phi_{3}$ can be obtained by evaluating at
$r=\sqrt{2\xi}$ a map that is analytic around $r=0$, exactly like
$\alpha_{3}(\vartheta,\xi)$ in (4.23), like $\gamma_{c}(\vartheta)$ in (4.36),
and like $\partial_{c}\gamma_{c}(\vartheta)$ in (4.51).

Similarly, both the map $\Phi_{4}$ in (3.52) and the map $\Phi_{5}$ in (3.63)
are analytic in $\mathcal{S}_{4}^{*}$, they are not analytic in $I$ around
$I=0$, but they can be obtained by evaluating at $\mu=\sqrt{I}$ some suitable
maps that are analytic around $\mu=0$, because
$\gamma_{c},F_{c},f_{c},g_{c},\sqrt{H(c)}$ are all functions of this type.

As a consequence, the map $\Phi$ in (3.73) is analytic in
$\mathcal{S}_{4}^{*}$, it is not analytic in $I$ around $I=0$, and it can be
obtained by evaluating at $\mu=\sqrt{I}$ a map that is analytic around
$\mu=0$. Hence $\Phi$ admits a converging expansion in powers of $\sqrt{I}$
around $I=0$.

###  4.9 Smallness conditions

The parameter $\delta$ in the definition (3.1) of the set $\mathcal{N}$ is
subject to the following smallness conditions. After (3.1) we have taken
$\delta\in(0,1)$ to obtain that $\mathcal{N}$ is an open neighborhood of the
circle $\mathcal{C}$ in (3.1) with $\rho>1-\delta>0$ in $\mathcal{N}$, and
$\delta\leq r_{0}$ to obtain that the functions $\alpha(\sqrt{x^{2}+y^{2}},z)$
and $H(\alpha(\sqrt{x^{2}+y^{2}},z))$ are analytic in $(x,y,z)\in\mathcal{N}$,
where $r_{0}$ is a universal constant given by the definition of $\alpha$ and
$H$, i.e., by Gavrilov’s construction in [7]. After (3.18) we have defined
$\delta_{2}=2\delta/3$, and we have assumed $\delta\leq 1/2$ to obtain the
inclusion (3.19). After (3.21) we have defined
$\xi_{3}=\delta_{2}^{2}/2=2\delta^{2}/9$. In (3.30) we have proved that
$\partial_{\xi}\alpha_{3}(\vartheta,0)=4$ for all $\vartheta\in{\mathbb{T}}$,
and that $\partial_{\xi}\alpha_{3}(\vartheta,\xi)$ is continuous in
${\mathbb{T}}\times[0,\xi_{3})$ — in fact, in Section 4.3 we have proved more,
because
$\alpha_{3}(\vartheta,\xi)=\widetilde{\alpha}_{3}(\vartheta,\sqrt{2\xi})$, see
(4.23), and $\widetilde{\alpha}_{3}(\vartheta,r)$ is the analytic function in
(4.15), (4.16); hence
$\partial_{\xi}\alpha_{3}(\vartheta,\xi)=\sum_{n=2}^{\infty}P_{n}(\vartheta)n(2\xi)^{(n-2)/2}=4+3P_{3}(\vartheta)\sqrt{2\xi}+O(\xi)$.
Hence, by continuity, there exists a universal constant $\xi^{*}>0$ such that
$\partial_{\xi}\alpha_{3}(\vartheta,\xi)>0$ in
${\mathbb{T}}\times[0,\xi^{*})$. Therefore the condition on $\xi_{3}$ after
(3.30) (where we say “Taking $\xi_{3}$ smaller if necessary”) is
$\xi_{3}\leq\xi^{*}$. In terms of $\delta$, this means $\delta\leq
3\sqrt{\xi^{*}/2}$, which is a universal constant.

The constants $w_{0},\mu_{0}$ after the definition (4.24) of $\mathcal{F}$ are
universal, and the constants $w_{1},\mu_{1}$ after the application of the
implicit function theorem in (4.26) are universal too. By (4.26),
$w(\vartheta,0)>0$, and therefore, by continuity, there exists a universal
constant $\mu_{1}^{*}>0$ such that $w(\vartheta,\mu)>0$ for all
$(\vartheta,\mu)\in{\mathbb{T}}\times(-\mu_{1}^{*},\mu_{1}^{*})$. Hence the
condition on $\mu_{1}$ after (4.34) (where we say “taking $\mu_{1}$ smaller if
necessary”) is $\mu_{1}\leq\mu_{1}^{*}$. The same happens for the condition on
$\mu_{1}$ after (4.54), to obtain that the function $m(\vartheta,\mu)$ in
(4.55) is well-defined and analytic in ${\mathbb{T}}\times(-\mu_{1},\mu_{1})$,
and for the condition on $\mu_{1}$ after (4.57), to obtain that $J(c)$ is
strictly increasing in $[0,\mu_{1}^{2})$. Thus, we fix $\mu_{1}$ as the
smallest of these three constants, and we obtain that $\mu_{1}$ is a universal
positive constant.

The parameter $\tau$ is related to $\mu_{1}$ by the inequality
$4\tau\leq\mu_{1}^{2}$, because $[0,4\tau)$ is the interval where $c$ varies,
and the functions $\gamma_{c}(\vartheta)$, $\partial_{c}\gamma_{c}(\vartheta)$
and $J(c)$ are obtained by evaluating at $\mu=\sqrt{c}$ some functions of
$(\vartheta,\mu)$ that are well-defined, analytic and monotonic in
${\mathbb{T}}\times(-\mu_{1},\mu_{1})$, see (4.36), (4.51), (4.57). Thus, we
want that, for all $c\in[0,4\tau)$, the square root $\sqrt{c}$ belongs to the
domain $(-\mu_{1},\mu_{1})$ of those analytic functions, and this is true if
$4\tau\leq\mu_{1}^{2}$.

Regarding the parameter $\tau$, there are two other conditions to consider.
The first one is (1.9), which holds if $\tau$ is smaller than the infimum of
the pressure $P$ on the set $\mathcal{N}\setminus\mathcal{N}^{\prime}$. Since
$P(x,y,z)=p(\rho,z)=\frac{1}{4}\alpha(\rho,z)$ (see (3.2)), by the definition
(3.1) of $\mathcal{N}$ and $\mathcal{N}^{\prime}$ it follows that that infimum
depends only on $\delta$. The last condition for $\tau$ is after (4.61), where
we say “Taking a smaller $I^{*}$ if necessary”, to obtain that
$\mathcal{R}(I)$ is strictly increasing in $[0,I^{*})$. Since the function
$\mathcal{R}$ does not depend on any parameter, this is a condition of the
form $I^{*}\leq I_{0}$ for some universal constant $I_{0}>0$. Moreover, the
invertible function $h_{1}$ expressing $I^{*}$ in terms of $\tau$ in (3.56) is
also independent on any parameter (see the definition of $h_{1}$ in (3.55) and
its expansion in (4.49)), and therefore this condition for $\tau$ is satisfied
for $\tau$ smaller than a universal constant.

Regarding $\varepsilon$, the only condition to consider is that
$0<\varepsilon\leq\tau/3$, see after (1.9).

In conclusion, the parameters $\delta,\tau$ and $\varepsilon$ must satisfy

$0<\delta\leq\delta_{0},\quad\ 0<\tau\leq\tau_{0}(\delta),\quad\
0<\varepsilon\leq\tau/3,$

where $\delta_{0}$ is a universal constant, and $\tau_{0}(\delta)$ depends
only on $\delta$. We fix $\delta=\delta_{0}$ and $\tau=\tau_{0}(\delta_{0})$.
Both $\delta_{0}$ and $\tau_{0}(\delta_{0})$ are universal constants. We
rename $\tau_{0}:=\tau_{0}(\delta_{0})$ and $\varepsilon_{0}:=\tau_{0}/3$.
Since $\tau=\tau_{0}$, by (3.56), $I^{*}$ is also a universal constant.

For notation convenience, we denote

$\mathcal{K}(I):=\frac{1}{4}h(I).$

Hence, by (3.67) and (3.6),
$\Omega_{1}(I)=\frac{1}{4}\omega(\frac{1}{4}h(I))h^{\prime}(I)=\omega(\mathcal{K}(I))\mathcal{K}^{\prime}(I)$;
by (4.58) and (4.48),

$\mathcal{K}(I)=\frac{1}{4}\Big{(}4I+256\frac{1065}{2^{16}}I^{3}+O(I^{4})\Big{)}=I+\frac{1065}{1024}I^{3}+O(I^{4});$

(3.76) becomes $P(\Phi(\sigma,\beta,I))=\mathcal{K}(I)$. For all
$\varepsilon\in(0,\varepsilon_{0}]$, the proof of Theorem 1.1 is complete.

Acknowledgements. Supported by the Project PRIN 2020XB3EFL _Hamiltonian and
dispersive PDEs_.

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Pietro Baldi

Dipartimento di Matematica e Applicazioni “R. Caccioppoli”

Università di Napoli Federico II

Via Cintia, 80126 Napoli, Italy

<EMAIL_ADDRESS>