# On the existence of periodic invariant curves for analytic families of twist
maps and billiards

Corentin Fierobe  Institute of Science and Technology Austria, Am Campus 1,
3400 Klosterneuburg, Austria<EMAIL_ADDRESS>and  Alfonso
Sorrentino  Department of Mathematics, University of Rome Tor Vergata, Via
della ricerca scientifica 1, 00133 Rome, Italy<EMAIL_ADDRESS>
###### Abstract.

In this paper we prove that in any analytic one-parameter family of twist maps
of the annulus, homotopically invariant curves filled with periodic points
corresponding to a given rotation number, either exist for all values of the
parameters or at most for a discrete subset. Moreover, we show that the set of
analytic twist maps having such an invariant curve of a given rotation number
is a strict analytic subset of the set of analytic twist maps. The first
result extends, in dimension $2$, a previous result by Arnaud, Massetti and
Sorrentino [4]. We then apply our result to rational caustics of billiards,
considering several models such as Birkhoff billiards, outer billiards and
symplectic billiards.

##  1\. Introduction

The study of invariant manifolds in Hamiltonian systems is a significant area
of research, stemming from the pioneering works of Poincaré and subsequent
seminal contributions by Kolmogorov, Arnold, and Moser, which laid the
groundwork for what is now known as KAM theory [21, 5, 28]). While the
investigation of the existence of invariant manifolds is crucial for
understanding the stability of these systems, their destruction holds pivotal
importance in elucidating the transition from stability to instability, from
integrable regimes to non-integrable ones.

In this article, we focus on twist maps of the annulus and the existence of
invariant curves. Of particular interest are invariant curves foliated by
periodic points, which emerge as delicate structures within the dynamical
landscape. Understanding the essence of this fragility plays a crucial role in
trying to tackle several of the foremost questions and conjectures in dynamics
(see for example, recent developments around integrable billiards and
Birkhoff’s conjecture [6, 17, 18, 22]).

More specifically, we consider real-analytic one-parameter families of exact
twist maps (see Definition 1 for a more precise statement) and investigate the
topological structure of the subset of parameters corresponding to maps
admitting an invariant curve foliated by periodic points and with a given
rational rotation number. Our main result can be summarized as follows:

Main Theorem 1 (see Theorem 8 for a more precise statement). Given an interval
$J$ and a real-analytic family of exact symplectic twist maps
$(F_{\varepsilon})_{\varepsilon\in J}$, the set of parameters $\varepsilon\in
J$ such that $F_{\varepsilon}$ admits an invariant curve filled with periodic
points of a given rotation number is either discrete or the whole interval
$J$.

This result generalizes to arbitrary real-analytic families of twist-maps of
the annulus a previous result by Arnaud, Massetti and Sorrentino [4], which
considered special families generated by perturbation by a potential, in any
dimension.

Actually, one could prove a more general version of Main Theorem 1 that does
not involve the parameter $\varepsilon$. In fact, one can show that the set of
twist maps having an invariant curve filled with periodic points of a given
rotation number has the structure of an analytic set; this means that it is
closed and it is locally given by the zeros of an analytic map.

Main Theorem 2 (see Theorem 12 for a more precise statement). The set of real-
analytic exact symplectic twist-maps having an invariant curve filled with
periodic points of a given rotation number is a strict analytic subset of the
set of real-analytic exact symplectic twist-maps.

Our main motivation for these results came from billiard dynamics. Therefore,
we apply the above theorems to several billiard models: classical Birkhoff
billiards, outer billiards and symplectic billiards.

The article is organized as follows:

  * •

In Section 3 we provide a more detailed description of the setting, so to
state more precisely our main theorems, see Theorems 8 and 12.
Moreover, we present the application of the main theorems to the various
billiard models; more specifically, Subsections 3.6 for classical Birkhoff
billiards, Subsection 3.7 for outer billiards and Subsection 3.8 for
symplectic billiards.

  * •

In Section 4 we recall and prove some properties of periodic invariant graphs.

  * •

Section 5 is devoted to the proof of the Main Theorem (Theorem 8).

  * •

In Sections 6, 7 and 8 we discuss the proofs of the results related to the
various billiard models.

  * •

Finally, in Section 9 we prove Main Theorem 2 and its application to Birkhoff
Billiards (Theorems 12 and 17).

##  2\. Acknowledgments

The authors are grateful to the Simons Center for Geometry and Physics at
Stony Brook for its hospitality and support during the program “Mathematical
billiards: at the crossroads dynamics, geometry, analysis, and mathematical
physics”, where part of this project was carried out.
AS also acknowledges the support of the Italian Ministry of University and
Research’s PRIN 2022 grant “Stability in Hamiltonian dynamics and beyond”, as
well as the Department of Excellence grant MatMod@TOV (2023-27) awarded to the
Department of Mathematics of University of Rome Tor Vergata. AS is a member of
the INdAM research group GNAMPA and the UMI group DinAmicI.

##  3\. Preliminaries and statements of the results

In this section we provide a more precise description of the setting and state
the main results and their applications to billiard models, in full details.

###  3.1. Exact-symplectic twist maps

On the space $\mathbb{R}^{2}$ of pairs $(p,q)$, consider
$\pi_{q},\pi_{p}:\mathbb{R}^{2}\to\mathbb{R}$ the projections, respectively,
onto $q$ and $p$.

Consider two continuous $\mathbb{Z}$-periodic maps
$p^{-},p^{+}:\mathbb{R}\to\mathbb{R}\cup\\{\pm\infty\\}$ and assume that the
inequality $p^{-}(q)<p^{+}(q)$ is satisfied for any $q\in\mathbb{R}$. We
define the open strip

$\mathbb{A}_{p^{\pm}}=\\{(q,p)\in\mathbb{R}^{2}\,|\,p^{-}(q)<p<p^{+}(q)\\}.$

It is a bundle over $\mathbb{R}$, whose fibers are the intervals

$\left.\mathbb{A}_{p^{\pm}}\right|_{q}=\\{p\in\mathbb{R}\,|\,p^{-}(q)<p<p^{+}(q)\\}.$

$\mathbb{A}_{p^{\pm}}$ projects onto an interval bundle over the torus
$\mathbb{T}^{1}:=\mathbb{R}/\mathbb{Z}$ having the same fibers. We denote by
$\text{graph}\,(p^{-})$ and $\text{graph}\,(p^{+})$, respectively, the graphs
of $p^{-}$ and $p^{+}$ in $\mathbb{R}^{2}$.

######  Definition 1.

A diffeomorphism $F:\mathbb{A}_{p^{\pm}}\longrightarrow\mathbb{A}_{p^{\pm}}$,
where $F(q,p):=(Q(q,p),P(q,p))$, is called an exact-symplectic twist map if it
satisfies the following properties:

  * (i)

(Periodicity) $F(q+m,p)=F(q,p)+(m,0)$ for any $(q,p)\in\mathbb{A}_{p^{\pm}}$
and $m\in\mathbb{Z}$;

  * (ii)

(Twist condition) for any $q\in\mathbb{R}$ the map

$p\in\left.\mathbb{A}_{p^{\pm}}\right|_{q}\longmapsto Q(q,p)$

is a diffeomorphism onto its image;

  * (iii)

(Boundary preservation) For any neighborhood $V$ of
$\text{graph}\,(p^{-})\cup\text{graph}\,(p^{+})$ in $\mathbb{R}^{2}$, there
exists another neighborhood $U$ satisfying

$F(U\cap\mathbb{A}_{p^{\pm}})\subseteq V\cap\mathbb{A}_{p^{\pm}};$

  * (iv)

(Generating function) There is an open set
$\mathcal{D}\subseteq\mathbb{R}^{2}$ and a smooth map
$S:{\mathcal{D}}\to\mathbb{R}$, called generating function of $F$, such that
for any $(q,Q)\in{\mathcal{D}}$ and $m\in\mathbb{Z}$:

$\displaystyle\quad(q,Q)\in{\mathcal{D}}\quad\Longrightarrow\quad(q+m,Q+m)\in{\mathcal{D}}$
$\displaystyle\quad S(p+m,Q+m)=S(q,Q),$

and

$PdQ-pdq=dS(q,Q).$

######  Remark 2.

Given an exact symplectic twist map $F$, the map
$p\in\left.\mathbb{A}_{p^{\pm}}\right|_{q}\mapsto Q(q,p)$ is either strictly
increasing for any $q$, or strictly decreasing for any $q$. In the first case,
we say that $F$ is positive, in the second one that it is negative.
Note also that since $\partial_{12}^{2}S=-(\partial_{p}Q)^{-1}$ [19, Formula
(9.2.4)], we observe that $\partial_{12}^{2}S<0$ if $F$ is positive, and
$\partial_{12}^{2}S>0$ if $F$ is negative. In the proofs, we will often assume
that $F$ is positive to simplify the redaction, since the proofs in the
negative case are analogous.

######  Remark 3.

An exact symplectic twist map $F$ induces a map $f$ from the tangent bundle
$T\mathbb{T}^{1}\simeq\mathbb{T}^{1}\times\mathbb{R}$ to itself. More
precisely, if $\pi:\mathbb{R}^{2}\to\mathbb{T}^{1}\times\mathbb{R}$ is the
canonical projection, then
$f:\pi(\mathbb{A}_{p^{\pm}})\to\pi(\mathbb{A}_{p^{\pm}})$ is defined by
$f\circ\pi=\pi\circ F.$ In order to ease notation, we will use the same
notations for both maps.

###  3.2. Periodic and invariant graphs

A rotational invariant curve of a symplectic twist map
$F:\mathbb{A}_{p^{\pm}}\to\mathbb{A}_{p^{\pm}}$ is a curve
$\Gamma\subset\mathbb{A}_{p^{\pm}}$ such that $F(\Gamma)=\Gamma$ and
$\mathbb{A}_{p^{\pm}}\setminus\Gamma$ consists of two connected components.

######  Remark 4.

A famous theorem by Birkhoff (see for example [11, Theorem 15.1]) states that
any rotational invariant curve $\Gamma$, if it exists, is the graph of a
Lipschitz continuous $1$-periodic map $\gamma:\mathbb{R}\to\mathbb{R}$.
Moreover, the Lipschitz constant of $\gamma$ only depends on
$\inf_{\Gamma}\partial_{p}Q>0$ (see [11, Theorem 15.1], [16, Lemma 13.1.1] and
[14, Proposition 12.3] for more details).

This leads to the following definition.

######  Definition 5.

Let $F:\mathbb{A}_{p^{\pm}}\to\mathbb{A}_{p^{\pm}}$ be an exact-symplectic
twist map, $\Gamma=\text{graph}(\gamma)\subset\mathbb{A}_{p^{\pm}}$ be the
graph of a $1$-periodic Lipschitz-continuous map
$\gamma:\mathbb{R}\to\mathbb{R}$, and $m\in\mathbb{Z},n\in\mathbb{N}^{*}$
coprime. We say that $\Gamma$ is

  * (i)

$(m,n)$-periodic if $F^{n}(q,\gamma(q))=(q+m,\gamma(q))$ for any
$q\in\mathbb{R}$;

  * (ii)

invariant by $F$ if $F(\Gamma)\subseteq\Gamma$;

  * (iii)

$\mathcal{C}^{k}$-smooth (resp. analytic) if $\gamma$ is a
$\mathcal{C}^{k}$-smooth (resp. analytic).

We show in Proposition 21 that $(m,n)$-periodic graphs are automatically
invariant by $F$ and have the same regularity as $F$.

###  3.3. Twist interval

Given an $(m,n)$-periodic invariant graph $\Gamma$ of $F$, the restrictions of
$\left.F\right|_{\Gamma}$ can be seen as the lift of a diffeomorphism of
$\mathbb{T}^{1}$ whose rotation number is $m/n$. We introduce the set of all
possible rotation numbers of orbits of $F$, and call it twist interval of $F$.
It is defined as follows (see also [16, Definition 9.3.2]):

######  Definition 6.

The twist interval of a symplectic twist map $F$ is the set $\text{TI}(F)$ of
numbers $\alpha\in\mathbb{R}$ for which there is a neighborhood $U^{-}$ of
$\text{graph}\,(p^{-})$ and a neighborhood $U^{+}$ of $\text{graph}\,(p^{+})$
in $\mathbb{R}^{2}$ such that for $(q,p)\in\mathbb{A}_{p^{\pm}}$

$(q,p)\in U^{-}\;\Longrightarrow\;\pi_{q}\circ F(q,p)-q\leq\alpha$

and

$(q,p)\in U^{+}\;\Longrightarrow\;\pi_{q}\circ F(q,p)-q\geq\alpha.$

######  Remark 7.

(i) The twist interval is by construction an open interval of $\mathbb{R}$.
For example, given $\varepsilon\in\mathbb{R}$, the map
$F_{\varepsilon}:\mathbb{R}\times(0,1)\to\mathbb{R}\times(0,1)$ defined for
all $(q,p)\in\mathbb{R}\times(0,1)$ by $F(q,p)=(q+p+\varepsilon,p)$ is an
exact symplectic twist map whose twist interval is
$\text{TI}(F_{\varepsilon})=(\varepsilon,1+\varepsilon).$
(ii) Another example can be given in the case of the usual billiard map (this
will discussed in more details in Section 3.6). Given a strictly convex domain
$\Omega$ (fix an orientation of its boundary $\partial\Omega$), the billiard
map $F_{\Omega}:\mathbb{R}\times(-1,1)\to\mathbb{R}\times(-1,1)$ is given by
$F_{\Omega}(q,-\cos\varphi)=(q_{1},-\cos\varphi_{1})$ where
$(q_{1},\varphi_{1})$ is the pair describing the point of impact and the angle
of reflection after the bounce of a trajectory coming from $q$ and making an
angle $\varphi$ with the (oriented) tangent vector at $\Omega$ in $q$. The
twist interval of $F_{\Omega}$ is given by $TI(F_{\Omega})=(0,1)$: indeed,
when $\varphi$ goes from $0$ to $\pi$, the point of impact $q_{1}$ moves along
the boundary $\partial\Omega$ from $q$ to itself, winding exactly once around
$\partial\Omega$.

###  3.4. Main theorem 1

We now consider analytic one-parameter families of symplectic twist maps. More
specifically, consider an interval $I\subset\mathbb{R}$ and continuous maps
$p^{-},p^{+}:I\times\mathbb{R}\to\mathbb{R}\cup\\{\pm\infty\\}$ that are
$1$-periodic in the second component and such that for any $(\varepsilon,q)\in
I\times\mathbb{R}$ the inequality $p^{-}(\varepsilon,q)<p^{+}(\varepsilon,q)$.
One can introduce the open set

$\mathbb{A}_{I,p^{\pm}}:=\\{(\varepsilon,q,p)\in
I\times\mathbb{R}^{2}\,|\,p^{-}(\varepsilon,q)<p<p^{+}(\varepsilon,q)\\}$

and denote its closure by

$\overline{\mathbb{A}}_{I,p^{\pm}}=\\{(\varepsilon,q,p)\in
I\times\mathbb{R}^{2}\,|\,p^{-}(\varepsilon,q)\leq p\leq
p^{+}(\varepsilon,q)\\}.$

Given $\varepsilon\in I$, we denote its $\varepsilon$-section by

$\mathbb{A}_{I,p^{\pm}}^{\varepsilon}:=\\{(q,p)\in\mathbb{R}^{2}\,|\,p^{-}(\varepsilon,q)<p<p^{+}(\varepsilon,q)\\}.$

######  Theorem 8 (Main Theorem 1).

Assume that $I\subset\mathbb{R}$ is an interval and
$(m,n)\in\mathbb{Z}\times\mathbb{N}^{\ast}$ are coprime. Suppose that for any
$\varepsilon\in I$ we are given an exact symplectic twist map
$F_{\varepsilon}$ such that:

  * (i)

the map $(\varepsilon,q,p)\in\mathbb{A}_{I,p^{\pm}}\mapsto
F_{\varepsilon}(q,p)$ is analytic;

  * (ii)

$m/n\in TI(F_{\varepsilon})$ for every $\varepsilon\in I$.

Then, the set

$\mathcal{I}_{(m,n)}(\mathbb{R}):=\left\\{\varepsilon\in
I\,|\,F_{\varepsilon}\text{ has an }(m,n)\text{-periodic invariant
graph}\right\\}$

is either discrete or consists of the whole $I$. In particular, if $I$ is
compact, then $\mathcal{I}_{(m,n)}(\mathbb{R})=I$ or it is at most finite.

######  Remark 9.

In the statement of Theorem 8, we do not need to precise the regularity of
invariant graphs. In fact, it follows from Proposition 21 that they are indeed
analytic.

###  3.5. Main Theorem 2

Theorem 8 can be generalized without involving the parameter $\varepsilon$.
Let us define the set $\mathscr{T}^{\omega}$ of analytic twist-maps of
$\mathbb{A}_{p^{\pm}}$ which admit a continuous extension to the closure
$\overline{\mathbb{A}_{p^{\pm}}}$. It is an open subset of the Banach set of
analytic maps of $\mathbb{A}_{p^{\pm}}$ to itself admitting a continuous
extension to $\overline{\mathbb{A}_{p^{\pm}}}$.

Before stating our result, let us recall the definition of analytic subset of
a Banach space. We refer the reader to [37] for more details about this topic.

######  Definition 10.

Let $U$ be an open set of a Banach space $E$ and let $X\subset U$ be a closed
subset.

  * •

$U$ is said an analytic subset of $U$ if for any $x\in U$, there is an
analytic map $f_{x}:V_{x}\to F$ defined on a neighborhood $V_{x}\subset U$ of
$x$ to a Banach space $F$ and such that $X\cap V_{x}=f_{x}^{-1}(0)$.

  * •

$X$ is said to be a strict analytic subset if $X\neq U$.

######  Remark 11.

If $U$ is connected, strict analytic subsets of $U$ must have empty interior.

We can now state our result.

######  Theorem 12 (Main Theorem 2).

Let $m/n\in\mathbb{Q}$. The subset of twist-maps of $\mathscr{T}^{\omega}$
having an $(m,n)$-periodic invariant graph is a strict analytic subset of
$\mathscr{T}^{\omega}$.

######  Remark 13.

One could actually deduce the proof of Theorem 8 using the statement of
Theorem 12. In fact, if $\varphi$ is the analytic map $\varphi:\varepsilon\in
I\mapsto F_{\varepsilon}\in\mathscr{T}^{\omega}$ and
$\mathscr{T}^{\omega}_{m,n}$ is the subset of twist-maps of
$\mathscr{T}^{\omega}$ having an $(m,n)$-periodic invariant graph, then
$\mathcal{I}_{(m,n)}(\mathbb{R})$ is simply
$\mathcal{I}_{(m,n)}(\mathbb{R})=\varphi^{-1}(\mathscr{T}^{\omega}_{m,n})$,
hence is an analytic subset of $I$. It follows that it is either the whole $I$
or it is discrete (this property on analytic subsets of $\mathbb{R}$ holds
also with our definition of analytic subset). However, we decided to provide
an independent proof of Theorem 8, which is closer to our initial motivations.

###  3.6. Application to Birkhoff billiards

A classical (planar) billiard is a bounded domain
$\Omega\subset\mathbb{R}^{2}$ with (piecewise) smooth boundary, in which one
can study the behaviour of an infinitely small particle evolving inside
$\Omega$ without friction. When reaching the boundary, the particle bounces on
it according to the usual reflection law of geometrical optics: the angle of
incidence equals the angle of reflection. We refer the interested reader to
[23, 35] for more details on billiards, and to [34, Chapter 3] for an overview
on rigidity questions in billiards.

$\Omega$ Figure 1. The classical reflection law of a particle inside a
strictly convex domain $\Omega$ with smooth boundary.

Let us consider the case when $\Omega$ is strictly convex with a smooth
boundary, which defines a so-called Birkhoff billiard. The dynamics of a
particle in $\Omega$ is described by a discrete map, the billiard map, acting
on the space of oriented lines intersecting $\Omega$: given a line $\ell$, the
billiard map associates to it the line $\ell^{\prime}$ naturally obtained by
reflecting $\ell$ at the point of impact with $\partial\Omega$ (see Figure 1).
The phase space is a cylinder, which can be parametrized by pairs
$(s,\varphi)\in\mathbb{R}/|\partial\Omega|\mathbb{Z}\times[0,\pi]$, where $s$
is an arc-length coordinate on the boundary $\partial\Omega$ and $\varphi$ is
the angle between the tangent line of $\Omega$ at $s$ and the corresponding
oriented line starting at $s$.

As for dynamical systems in general, one can study the so-called integrable
billiards: billiards whose phase space contains an open set foliated by curves
which are invariant by the billiard map. Circles are examples of such
billiards for which the whole phase space is foliated by invariant curves – in
this case we speak about globally integrable billiards. Let us mention also
the case of ellipses, which are integrable, but not globally integrable. A
famous conjecture, due to Birkhoff [8] and Poritsky [31], states that

######  Conjecture 14 (Birkhoff-Poritsky).

The only integrable billiards are ellipses.

Bialy [6] showed that the only globally integrable billiards are circles.
Kaloshin-Sorrentino [18] proved that the only billiards close to ellipses
having invariant curves of rotation number $1/q$ for any $q\geq 2$ are
ellipses. Later, Koval [22] extended this result to billiards close to generic
ellipses having invariant curves of rotation number $r=p/q$, for any $r$ lower
than an arbitrarily small bound of the form $1/q_{0}$. The rotation number of
an invariant curve is defined as the rotation number of the circle map
obtained by restricting the billiard map to the corresponding curve.

In the case of non-rational rotation number, Lazutkin [24] showed that there
is a Cantor set $C\subset[0,1]$ of non-zero measure accumulating to $0$ such
that each $\omega\in C$ is the rotation number of an invariant curve. Popov
showed [30] that these curve persists under a small deformation of the
billiard. However the rotation numbers considered in these results are far
from being rational: they are so-called Diophantine numbers, which are numbers
badly approximated by rationals.

In the case of rational rotation numbers, it is expected that corresponding
invariant curves are more fragile.

The main result in this paper allows us to prove the following result.

######  Theorem 15 (Invariant curves in families of Birkhoff billiards).

Let $I$ be an interval and $(\Omega_{\varepsilon})_{\varepsilon\in I}$ be an
analytic family of strictly convex analytic domains. Then given a pair
$(m,n)\in\mathbb{Z}\times\mathbb{N}^{\ast}$ of coprime integers such that
$m/n\in(0,1)$, the set of $\varepsilon\in I$ such that the billiard map inside
$\Omega_{\varepsilon}$ has an $(m,n)$-periodic invariant curve is either
discrete or consists of the whole $I$.

This result does not answer Birkhoff’s conjecture, but rather confirms how
fragile integrability is. Let us also mention the work of Kaloshin-Koudjinan
[17] in which they study for billiard domains close to a disk the coexistence
of two invariant curves of rotation numbers $1/2$ and $1/2q+1$ for $q\geq 1$.

The proof Theorem 15 is a consequence of Main Theorem (Theorem 8), as it will
be shown in Section 6.

######  Remark 16.

An interesting application of Theorem 15 regards analytic families of Birkhoff
billiards generated by the action of a geometric flow. One of the main
motivations comes from [18, Appendix F], where the authors proposed a possible
approach to extend their local analysis (near ellipses) to a more global one,
by considering the evolution of a given Birkhoff billiard under the so-called
affine curvature flow: the family of billiards thus obtained will “converge”
(up to renormalize their area) to ellipses (see for example [33] for more
details).

More precisely, given a strictly convex bounded domain $\Omega$ with analytic
boundary and $\delta>0$, there is a one parameter family
$(\Omega_{\varepsilon})_{\varepsilon\in[0,T]}$ of strictly convex bounded
domains satisfying the following properties:

  * •

$\Omega_{0}=\Omega$ and $\Omega_{T}$ is $\delta$-close to an ellipse;

  * •

for $\varepsilon>0$, the domain $\Omega_{\varepsilon}$ has an analytic
boundary, and the family $(\Omega_{\varepsilon})_{\varepsilon\in(0,T]}$ is
analytic in $\varepsilon$.

The crucial observation is that $\Omega_{\varepsilon}$ are ellipses if and
only if one of them is an ellipse.
One of the main obstacle in applying this idea is that it is not easy to show
that integrability is preserved by the action of the flow: this statement
turns out to be equivalent to Birkhoff conjecture.
An immediate application of Theorem 15 implies that either the family consists
of all integral billiards $(\Omega_{\varepsilon})_{\varepsilon\in[0,T]}$ or
there are at most finitely many elements of the family that are integrable.

Following the same approach as in Theorem 12, we can generalize the statement
of Theorem 15 in the following way.

In the set of domains with analytic boundary, we distinguish those whose
boundary has a strip of analyticity of at least a given size, so to be able to
consider Banach spaces of domains.

It is well-known that, up to translations and rotations, strictly convex
domains of $\mathbb{R}^{2}$ with $\mathscr{C}^{k+1}$-smooth boundary is in
one-to-one correspondence with the set of $\mathscr{C}^{k}$-smooth
$1$-periodic maps $\varrho:\mathbb{R}\to\mathbb{R}^{>0}$ satisfying

(1) $\int_{0}^{1}\varrho(\theta)e^{2i\pi\theta}d\theta=0.$

The idea is that given such a map $\varrho$, the function $\gamma$ defined for
any $\theta\in[0,2\pi]$ by

$\gamma(\theta):=\left(\int_{0}^{\theta}\varrho(t)\cos(t)dt,\int_{0}^{\theta}\varrho(t)\sin(t)dt\right)$

parametrizes a simple closed curve containing the origin and which is tangent
to the $x$-axis at the origin ($\theta$ represents the angle that the tanget
to the curve forms with the positive $x$ semi-axis). This idea is developped
in more details in [27, Section 2].

Hence, given $r>0$ we consider the set $\mathscr{D}_{r}$ of $1$-periodic
analytic maps $\varrho:\mathbb{R}\to\mathbb{R}^{>0}$ satisfying (1) and having
a strip of analyticity of size at least $r$, in the following sense: if the
Fourier expansion of $\varrho$ is
$\varrho(\theta):=\sum_{p\in\mathbb{Z}}\widehat{\varrho}_{p}e^{2i\pi
p\theta}$, then we assume that $\widehat{\varrho}_{p}=\mathcal{O}(e^{-r|p|})$
as $p\to\pm\infty$. $\mathscr{D}_{r}$ is a Banach space. Thus we can consider
analytic subsets of $\mathscr{D}_{r}$, as introduced in Definition 10 and
prove the following theorem.

######  Theorem 17.

Let $m/n\in(0,1)$. The set of domains $\Omega\in\mathscr{D}_{r}$ such that the
billiard map in $\Omega$ has an $(m,n)$-periodic invariant graph is a strict
analytic subset of $\mathscr{D}_{r}$.

######  Remark 18.

Similarly to what observed in Remark 13, one could use Theorem 17 to obtain an
alternative proof of Theorem 15.

###  3.7. Application to dual billiards

Given a strictly convex domain $\Omega\subset\mathbb{R}^{2}$ with a smooth
oriented boundary, the dual or outer billiard outside $\Omega$ can be defined
as follows (see Figure 2). For any point $p\in\mathbb{R}^{2}\setminus\Omega$,
there are at most two tangent lines to $\partial\Omega$ passing through $p$.
Consider the unique one which is tangent to $\partial\Omega$ at a point $q$
and such that the vector $\vec{pq}$ has the same orientation as the boundary
$\partial\Omega$ at $q$. Define the image by $p$ by the dual billiard map as
the point $F(p)$ on the latter tangent line $T_{q}\partial\Omega$ such that
$q$ is the midpoint between $p$ and $F(p)$ (see Figure 2).

$\Omega$$q$$F(p)$$p$ Figure 2. The point $F(p)$ is the image of $p$ by the
dual billiard map around the domain $\Omega$. The point $q$ is the midpoint
between $p$ and $F(p)$ which are supported by a line tangent to
$\partial\Omega$ at $q$.

According to [29], dual billiards were introduced by B. H. Newman in 1960 and
also mentionned by P. C. Hammer in [15] a bit later. Their properties were
largely studied since then. They are known to be symplectic twist maps of the
infinite annulus $\mathbb{T}^{1}\times(0,+\infty)$, and thus can be
investigated in the context of Aubry-Mather theory [9]. Douady [10] showed
that if the boundary $\partial\Omega$ is sufficiently smooth (at least
$\mathcal{C}^{6}$), then there is a positive measure set of invariant curves
accumulating to the boundary, as well as a positive measure set of invariant
curves accumulating at infinity. In paticular this gives a negative answer to
the famous question of the existence of unbounded orbits. Adapting a result of
Mather for Birkhoff billiards (see [11]) Boyland [9] proved that if the
curvature of a convex domain vanishes, or has jump discontinuities, then there
is a neighborhood of the boundary without invariant curves.

A version of Birkhoff’s conjecture is also studied for dual billiards. It is
indeed known that the phase space of dual billiards around ellipses is
foliated by invariant curves, induced by any bigger ellipse homothetically
equivalent to the initial billiard. Bialy [7] proved a total integrability
result: if the phase space of a dual billiard is foliated by continuous
invariant curves then the billiard is an ellipse. If we assume the foliation
to be only in a open set of the phase space (local integrability), some
partial positive results were given in [13, 36].

In this context, our main result reads:

######  Theorem 19 (Invariant curves in families of outer billiards).

Let $I$ be an interval and $(\Omega_{\varepsilon})_{\varepsilon\in I}$ be an
analytic family of strictly convex analytic domains. Then given a pair
$(m,n)\in\mathbb{Z}\times\mathbb{N}^{\ast}$ of coprime integers such that
$m/n\in(0,1)$, the set of $\varepsilon\in I$ such that the outer billiard map
associated to $\Omega_{\varepsilon}$ has an $(m,n)$-periodic invariant curve
is either discrete or consists of the whole $I$.

Theorem 19 for outer billiards is a consequence of Main Theorem (Theorem 8),
as it will be shown in Section 7.

###  3.8. Application to symplectic billiards

As for classical billiards, symplectic billiards – which were firstly
introduced by Albers and Tabachnikov in [1] – are defined inside a strictly
convex bounded domain $\Omega$ with smooth boundary. They describe the
evolution of a infinitesimally small ball inside $\Omega$ which bounces on the
boundary $\partial\Omega$ according to the following reflection law: given
three successive impact points $p_{1}$, $p_{2}$ and $p_{3}$, the line joining
$p_{1}p_{3}$ and the tangent line $T_{p_{2}}\partial\Omega$ of
$\partial\Omega$ at $p_{2}$ are parallel (see Figure 3). We refer to [1] for
more details and results related to this billiard model.

figsimplbill $p_{1}$$p_{2}$$p_{3}$$T_{p_{2}}\partial\Omega$$\Omega$

Figure 3. A symplectic billiard bounce $(p_{1},p_{3})\mapsto(p_{2},p_{3})$ in
a strictly convex domain $\Omega$.

The symplectic billiard map $(p_{1},p_{2})\mapsto(p_{2},p_{3})$ describes the
symplectic billiard trajectories inside $\Omega$. It appears [1] that there
exists a set of coordinates in which this map can be expressed as an exact
symplectic twist map whose generating map is related to the canoncial
symplectic area form in the plane of coordinates $(x,y)$, namely $dx\wedge dy$
\- whence the name of this billiard model. It was shown, see [1, Theorem 3],
that the latter map has a positive measure set of invariant curves, which in
fact are caustics. The latter however have highly irrational rotation numbers,
and are not concerned by our result.

Still, we show that Theorem 8 can be applied to rational invariant curves in
analytic families of symplectic billiards; namely:

######  Theorem 20 (Invariant curves in families of symplectic billiards).

Let $I$ be an interval and $(\Omega_{\varepsilon})_{\varepsilon\in I}$ be an
analytic family of strictly convex analytic domains. Then given a pair
$(m,n)\in\mathbb{Z}\times\mathbb{N}^{\ast}$ of coprime integers such that
$m/n\in(0,1)$, the set of $\varepsilon\in I$ such that the symplectic billiard
map associated to $\Omega_{\varepsilon}$ has an $(m,n)$-periodic invariant
curve is either finite or consists of the whole $I$.

###  3.9. Other billiard models

In previous subsections, we described how Theorem 8 can be applied to
different billiard models such as the classical inner billiards, outer
billiards and symplectic billiards. Actually Theorem 8 could be applied to
many other billiard models, as long as their dynamics can be encoded by an
exact-symplectic twist map.

For a list of billiard models, the interested reader can consult [2]. As an
example, let us mention the outer length billiard, which is an analogue of the
outer billiard model, but with a different generating function. In fact,
periodic orbits are given by extremizing not the area of a circumscribed
polygon – as for outer billiards, but its perimeter. In this setting, the two
pairs (classical billiards, symplectic billiards) and (outer length billiards,
outer billiards) can be seen as “dual” to each other.

##  4\. Preliminary results on periodic graphs

In this section we prove some preliminary results on invariant periodic graphs
of a symplectic twist map $F:\mathbb{A}_{p^{\pm}}\to\mathbb{A}_{p^{\pm}}$ with
generating function $S:\mathcal{D}\to\mathbb{R}$. The main results in this
section are Proposition 21 and Proposition 23.

We first recall some basic notions related to the orbits of $F$, and we refer
the reader to [14] for more details about them. The projection
$\pi_{q}:\mathbb{R}^{2}\to\mathbb{R}$ onto the $q$-component induces a
bijection between orbits $(q_{k},p_{k})_{k\in\mathbb{Z}}$ of $F$, where for
all $k\in\mathbb{Z}$

$F(q_{k},p_{k})=(q_{k+1},p_{k+1}),$

and so-called stationary configurations $(q_{k})_{k\in\mathbb{Z}}$, which are
sequences satisfying for all $k\in\mathbb{Z}$

$(q_{k},q_{k+1})\in\mathcal{D}\qquad\text{and}\qquad\partial_{2}S(q_{k-1},q_{k})+\partial_{1}S(q_{k},q_{k+1})=0.$

An orbit $(q_{k},p_{k})_{k\in\mathbb{Z}}$ or its associated stationary
configuration $(q_{k})_{k\in\mathbb{Z}}$ are called minimal if for any
integers $u\leq v$ the family $(q_{k})_{u\leq k\leq v}$ minimizes the
functionnal

$\underline{x}=(x_{k})_{u\leq k\leq
v}\longmapsto\sum_{k=u}^{v-1}S(x_{k},x_{k+1})$

among all families $\underline{x}\in\mathbb{R}^{v-u+1}$ with $x_{u}=q_{u}$ and
$x_{v}=q_{v}$.

It is known (see [3, Proposition 6] or [6, Proof of Theorem A] or), that a
minimal orbit $(q_{k},p_{k})_{k\in\mathbb{Z}}$ has no conjugate points, which
means that any two points $(q_{k},p_{k})$ and
$(q_{\ell},p_{\ell})=F^{\ell-k}(q_{k},p_{k})$ along the orbit satisfy

$\partial_{p}\left(\pi_{q}\circ F^{\ell-k}\right)(q_{k},p_{k})\neq 0.$

######  Proposition 21.

Let $F:\mathbb{A}_{p^{\pm}}\longrightarrow\mathbb{A}_{p^{\pm}}$ be an exact-
symplectic twist map and $\Gamma\subset\mathbb{A}_{p^{\pm}}$ be an
$(m,n)$-periodic Lipschitz continuous graph of $F$, for some
$m\in\mathbb{Z},n\in\mathbb{N}^{*}$ coprime. Then:

  * (i)

$\Gamma$ is invariant by $F$;

  * (ii)

the projection of an orbit intersecting $\Gamma$ is a minimal configuration,
and hence has no conjugate points;

  * (iii)

$\Gamma$ is as smooth as $F$ is;

  * (iv)

$F$ has no other $(m,n)$-periodic Lipschitz continuous graph.

######  Remark 22.

Note that whereas there can exist at most one invariant rotational curve for
each irrational rotation number in the twist interval (see for instance [19,
Theorem 13.2.9]), this is not the case for rational rotation number. In fact,
two invariant graphs with the same rational rotation number might coexist: in
this case, their intersection consists of periodic orbits and the two graphs
should contain also non-periodic ones. For instance, consider the twist map
corresponding to an elliptic billiard (non-circular): its phase space contains
two invariant graphs of rotation number $1/2$.

###### Proof.

It is well-known that Items (i) and (ii) are equivalent, see for example [11,
Theorem 17.4] for the implication (i) $\Rightarrow$ (ii), and [4, Proposition
2.5] for the reverse implication. So to prove the result it is enough to show
that items (i), (iii) and (iv) are satisfied.

(i) To show that $\Gamma$ is invariant by $F$, we extend $F$ to a symplectic
twist map $G:\mathbb{R}^{2}\to\mathbb{R}^{2}$, coinciding with $F$ in a
neighborhood of $\Gamma$ and satisfying the superlinearity condition at
infinity, namely

$\lim_{|q-Q|\to+\infty}\frac{S_{G}(q,Q)}{|Q-q|}=+\infty,$

where $S_{G}$ denotes the generating function of $G$. If such a $G$ exists,
then the orbits of $G$ intersecting $\Gamma$ are minimal by [4, Proposition
2.5]; therefore, $\Gamma$ is invariant by $G$, and hence by $F$.

The construction of such a $G$ is quite standard, and we refer to [11, Section
8] or [14, Lemma 8.2]. It is however not necessarily analytic, but this does
not affect the result. Let us assume that $F$ is positive, and let $S$ be the
generating function of $F$: it is defined on an open set $\mathcal{D}$.
Consider the set of pairs $(q,Q)\in\mathcal{D}$ such that there is a point
$x\in\Gamma$ and an integer $k\in\\{0,\dots,n-1\\}$ for which $\pi_{q}\circ
F^{k}(x)=q$ and $\pi_{q}\circ F^{k+1}(x)=Q$; this set is contained in a
compact set $K\subset\mathcal{D}$. On this compact set the twist condition is
uniform, which means that there is $a>0$ such that
$\partial_{12}^{2}S_{|K}<-a$. Hence applying [11, Section 8], or [14, Lemma
8.2] we deduce the existence of a symplectic twist map
$G:\mathbb{R}^{2}\longrightarrow\mathbb{R}^{2}$ whose generating function
$S_{G}$ is defined on $\mathbb{R}^{2}$, coincides with $S$ on $K$ and has the
uniform twist everywhere, that is $\partial_{12}^{2}S_{G}<-a^{\prime}$
everywhere, for some $a^{\prime}>0$. This implies together with [14,
Proposition 11.2] that the map $G$ has the announced properties, which
concludes the result.

(iii) The smoothness comes from the property that an orbit corresponding to an
action-minimizing configuration has no conjugate points (see the proof of [6,
Theorem A] or [3, Proposition 6]). Hence following the argument of [4,
Proposition 2.5], this implies that the map $R:(q,p)\mapsto\pi_{q}\circ
F^{n}(q,p)-q-m$ which vanishes on $\Gamma$ satisfies $\partial_{p}R(q,p)\neq
0$ for $(q,p)\in\Gamma$. By the implicit mapping theorem, $\Gamma$ can be
described locally as a graph of a map which is as smooth as $F$, which proves
the assertion.

(iv) We apply [16, Lemma 13.2.10]: if $F$ has an invariant curve $\Gamma$ of
rotation number $m/n$, then any order-preserving orbit of $F$ whose closure is
distinct from $\Gamma$ has rotation number $\neq m/n$ (the definition of
order-preserving orbit is given in [16], and periodic orbits are a particular
case of them). Hence any periodic orbit of rotation number $m/n$ should have a
point on $\Gamma$, and consequently they must be entirely contained in
$\Gamma$ since $\Gamma$ is invariant by $F$. ∎

Let us show a localisation result when one consider a family of symplectic
twist maps.

######  Proposition 23.

Assume that $I\subset\mathbb{R}$ is a compact interval and
$(m,n)\in\mathbb{Z}\times\mathbb{N}^{\ast}$. Suppose that for any
$\varepsilon\in I$ we are given an exact symplectic twist map
$F_{\varepsilon}$ such that:

  * (i)

the map $(\varepsilon,q,p)\in\mathbb{A}_{I,p^{\pm}}\longmapsto
F_{\varepsilon}(q,p)$ is $\mathcal{C}^{1}$-smooth;

  * (ii)

$m/n\in TI(F_{\varepsilon})$ for any $\varepsilon\in I$.

Then, there is a neighborhood $U$ of
$\text{graph}\,(p^{-})\cup\text{graph}\,(p^{+})$ in
$I\times\mathbb{R}\times\mathbb{R}$ such that for any $\varepsilon\in I$, any
$(m,n)$-periodic orbit of $F_{\varepsilon}$ is contained in
$K:=\mathbb{A}_{I,p^{\pm}}\setminus U$.

From this result together with Remark 4, we immediately deduce the following
result.

######  Corollary 24.

Under the assumption of Proposition 23, there is a constant $k>0$ depending
only on $\inf_{K}\partial_{p}(\pi_{q}\circ F)$ such that for any
$\varepsilon\in I$, an $(m,n)$-periodic invariant graph $\Gamma$ of
$F_{\varepsilon}$ is Lipschitz continuous with Lipschitz constant $k$.

In order to prove Proposition 23, we need the following Lemma, whose proof can
be found in [19, Theorem 9.3.7 and its proof].

######  Lemma 25.

Let $F$ be an exact-symplectic twist map of the open annulus
$\mathbb{A}_{p^{\pm}}$ and $m/n\in TI(F)$. Then, given $a,b\in TI(F)$ such
that $a<m/n<b$, there exists a neighborhood $U_{-}$ of $\text{graph}\,(p^{-})$
and a neighborhood $U_{+}$ of $\text{graph}\,(p^{+})$ such that for any
$k\in\\{0,\ldots,n-1\\}$

(2) $\forall(q,p)\in U_{-}\qquad\pi_{q}\circ F^{k+1}(q,p)-\pi_{q}\circ
F^{k}(q,p)<a$

and

(3) $\forall(q,p)\in U_{+}\qquad\pi_{q}\circ F^{k+1}(q,p)-\pi_{q}\circ
F^{k}(q,p)>b.$

We can now prove Proposition 23.

###### Proof of Proposition 23.

Consider a decreasing sequence $U_{j}$ of neighborhoods of
$\text{graph}\,(p^{-})\cup\text{graph}\,(p^{+})$ in
$I\times\mathbb{R}\times\mathbb{R}$ such that
$\cap_{j}U_{j}=\text{graph}\,(p^{-})\cup\text{graph}\,(p^{+})$ and assume by
contradiction that there are two sequences $(\underline{x}^{(j)})_{j}$ and
$(\varepsilon_{j})_{j}$ such that for each $j$, $\underline{x}^{(j)}$ is an
$(m,n)$-periodic orbits of $F_{\varepsilon_{j}}$ having a point in
$(\\{\varepsilon_{j}\\}\times\mathbb{A}_{I,p^{\pm}}^{\varepsilon_{j}})\cap
U_{j}$.

Since $I$ is compact, we can assume that the sequence $(\varepsilon_{j})_{j}$
converges to a certain $\varepsilon\in I$. Consider the sets $U_{-}$ and
$U_{+}$ of Lemma 25 associated to the rotation number $m/n$ and the twist map
$F_{\varepsilon}$. Define $U=U_{-}\cup U_{+}$. By continuity of $F$, $p^{-}$
and $p^{-}$, for $\varepsilon^{\prime}$ sufficiently close to $\varepsilon$,
$U$ remains a neighborhood of
$\text{graph}\,(p^{-})(\varepsilon^{\prime},\cdot)\cup\text{graph}\,(p^{+})(\varepsilon^{\prime},\cdot)$
and Equations (2) and (3) are also satisfied by $F_{\varepsilon^{\prime}}$. In
particular, considering $\varepsilon^{\prime}=\varepsilon_{j}$ for
sufficiently large $j$, $(m,n)$-periodic orbits of $F_{\varepsilon_{j}}$ have
their points in $\mathbb{A}_{I,p^{\pm}}^{\varepsilon_{j}}\setminus U$. Now by
the assumptions made on the sequence of $U_{j}$’s, if $j$ is sufficiently
large $U_{j}\subset I\times U$, and the latter implies that
$\underline{x}^{(j)}$ is an $(m,n)$-periodic orbit of $F_{\varepsilon_{j}}$
having a point in $U$. This is contradictory and concludes the proof. ∎

##  5\. Proof of Main Theorem 1 (Theorem 8)

In this section, we prove Main Theorem 1, namely Theorem 8. Given a pair
$(m,n)\in\mathbb{Z}\times\mathbb{N}^{*}$ of coprime integers, we first recall
some properties of the set

$\mathcal{I}_{(m,n)}(\mathbb{R})=\\{\varepsilon\in I\,|\,F_{\varepsilon}\text{
has an }(m,n)\text{-periodic invariant graph}\\}$

defined in the statement of the theorem.

######  Lemma 26.

Under the assumptions of Theorem 8.

  * (i)

For $\varepsilon\in\mathcal{I}_{(m,n)}(\mathbb{R})$, there exists a unique
$1$-periodic continuous map $\gamma_{\varepsilon}:\mathbb{R}\to\mathbb{R}$
such that $\text{graph}(\gamma_{\varepsilon})$ is an $(m,n)$-periodic graph
invariant by $F_{\varepsilon}$.

  * (ii)

The map
$(\varepsilon,q)\in\mathcal{I}_{(m,n)}(\mathbb{R})\times\mathbb{R}\longmapsto\gamma_{\varepsilon}(q)$
is continuous (for the topology induced by $I\times\mathbb{R}$ on
$\mathcal{I}_{(m,n)}(\mathbb{R})\times\mathbb{R}$).

  * (iii)

$\mathcal{I}_{(m,n)}(\mathbb{R})$ is a closed subset of $I$.

###### Proof.

(i) The existence of such $\gamma_{\varepsilon}$ follows from the definition
of $\mathcal{I}_{(m,n)}(\mathbb{R})$. The unicity comes from Proposition 21.
Note that $\gamma_{\varepsilon}$ is Lipschitz continuous with a Lipschitz
constant depending on $\varepsilon$, but which can be chosen uniformly for any
$\varepsilon$ lying in a compact subinterval $I^{\prime}\subset I$ by
Corollary 24. This property will be useful for the rest of the proof.

(ii) Choose $\varepsilon\in\mathcal{I}_{(m,n)}(\mathbb{R})$ and a sequence
$(\varepsilon_{j})_{j}\subset\mathcal{I}_{(m,n)}(\mathbb{R})$ converging to
$\varepsilon$. Let us show that $(\gamma_{\varepsilon_{j}})_{j}$ converges to
$\gamma_{\varepsilon}$ in the uniform topology. As noticed in item (i), each
$\gamma_{\varepsilon_{j}}$ as well as $\gamma_{\varepsilon}$ is Lipschitz
continuous, and they share the same Lipschitz constant (i.e., they are equi-
Lipschitz).

We do the proof in two steps:

$\bullet$ Step 1. We prove that any subsequence of
$(\gamma_{\varepsilon_{j}})_{j}$ converging in the space of $1$-periodic
continuous maps $\mathbb{R}\to\mathbb{R}$ for the uniform topology has
$\gamma_{\varepsilon}$ as a limit.

$\bullet$ Step 2. We show that $(\gamma_{\varepsilon_{j}})_{j}$ has at least
one converging subsequence (applying Ascoli-Arzelà theorem).

These two steps imply the assertion of item (ii).

Proof of Step 1. Assume that there is a subsequence
$(\gamma_{\varepsilon_{j_{k}}})_{k}$ converging to a $1$-periodic map
$\gamma_{\infty}:\mathbb{R}\to\mathbb{R}$ in the uniform topology. By
Proposition 23,
$\text{graph}\,({\gamma_{\infty}})\subset\mathbb{A}_{I,p^{\pm}}^{\varepsilon}$.
The identity $F^{n}_{\varepsilon_{j_{k}}}=\text{Id}+(m,0)$ is satisfied on
$\text{graph}\,\gamma_{\varepsilon_{j_{k}}}$, hence, by continuity,
$F^{n}_{\varepsilon}=\text{Id}+(m,0)$ on $\text{graph}\,({\gamma_{\infty}})$.
Let us mention that $\gamma_{\infty}$ is Lipschitz continuous since all
$\gamma_{\varepsilon_{j_{k}}}$ are Lipschitz continuous with the same
Lipschitz constant. Hence $\text{graph}\,\gamma_{\infty}$ is a
$(m,n)$-periodic graph of $F_{\varepsilon}$, which is invariant by Proposition
21. Therefore $\gamma_{\infty}=\gamma_{\varepsilon}$, which follows from the
unicity of $(m,n)$-periodic graphs, again by Proposition 21.

Proof of Step 2. Let us check that the assumptions of Ascoli-Arzelà theorem
are satisfied by the sequence of maps $(\gamma_{\varepsilon_{j}})_{j}$. By
Proposition 23, the maps $\gamma_{\varepsilon_{j}}$ are contained in a compact
subset $K$ of $\mathbb{A}_{I,p^{\pm}}$ which implies that they are bounded by
the same constant (i.e., equibounded). Moreover, we have already noticed that
they are Lipshitz continuous with a uniform Lipschitz constant. Hence Step 2
follows from Ascoli-Arzelà theorem, and therefore point (ii) is proven.

(iii) Consider a sequence of
$\varepsilon_{j}\in\mathcal{I}_{(m,n)}(\mathbb{R})$ converging to a
$\varepsilon\in I$. As in item (ii), the family of maps
$\gamma_{\varepsilon_{j}}$ is equibounded and equi-Lipschitz. By extracting a
subsequence as in Step 2 of item (ii), we can suppose that
$\gamma_{\varepsilon_{j}}$ converges to a $1$-periodic Lipschitz continuous
map ${\gamma}_{\varepsilon}:\mathbb{R}\to\mathbb{R}$, such that
$\\{\varepsilon\\}\times\text{graph}\,(\gamma_{\varepsilon})\subset\mathbb{A}_{I,p^{\pm}}$
(Proposition 23), which is $(m,n)$-periodic by continuity of $F$ as a map of
$(\varepsilon,q,p)$. By Proposition 21, this implies that
$\varepsilon\in\mathcal{I}_{(m,n)}(\mathbb{R})$.
∎

We can now prove the following stronger result on the topological structure of
the set $\mathcal{I}_{(m,n)}(\mathbb{R})$.

######  Lemma 27.

Under the assumptions of Theorem 8, the set $\mathcal{I}_{(m,n)}(\mathbb{R})$
is either the whole $I$ or it has empty interior.

###### Proof.

Assume that $\mathcal{I}_{(m,n)}(\mathbb{R})$ has non-empty interior, and let
us show that $\mathcal{I}_{(m,n)}(\mathbb{R})=I$. Let us define $a:=\inf I$
and $b:=\sup I$, hence $I\cap(a,b)=(a,b)$.

Consider a connected component $A\subset\mathcal{I}_{(m,n)}(\mathbb{R})$ which
is not reduced to a point. Let $\beta=\sup A$, and suppose that $\beta<b$. We
will raise a contradiction.

First note that necessarily $\beta\in\mathcal{I}_{(m,n)}(\mathbb{R})$ since
$\beta\in I$ and the set $\mathcal{I}_{(m,n)}(\mathbb{R})$ is closed in $I$ by
Lemma 26; therefore, $\beta\in A$.

Applying again Lemma 26, we can then define a family of Lipschitz continuous
maps $(\gamma_{\varepsilon})_{\varepsilon\in A}$ such that for all
$\varepsilon\in A$, $\text{graph}(\gamma_{\varepsilon})$ is an
$(m,n)$-periodic graph invariant by $F_{\varepsilon}$. Moreover the map
$\Gamma:(\varepsilon,q)\mapsto\gamma_{\varepsilon}(q)$ is continuous, again by
Lemma 26. We will show that we can extend $\Gamma$ to the open set
$(\beta-r,\beta+r)\times\mathbb{R}$, with $r>0$, thus leading to a
contradiction, since this would imply that $(\beta-r,\beta+r)\subseteq A$,
thus contradicting the maximality of $A$.

Let us apply the implicit function theorem to the map

$\Delta_{1}(\varepsilon,q,p):=\pi_{q}\circ F^{n}_{\varepsilon}(q,p)-q-m.$

Since $F_{\beta}$ has no conjugate points on $\text{graph}(\gamma_{\beta})$
(see Proposition 21), $\partial_{p}\Delta_{1}$ do not vanish on the set
$\\{(\beta,q,\gamma_{\beta}(q))\,|\,q\in\mathbb{R}\\}$. Hence we can define an
analytic map
$(\varepsilon,q)\in(\beta-r,\beta+r)\times\mathbb{R}\mapsto\eta_{\varepsilon}(q)$,
with $r>0$, such that $\eta_{\beta}=\gamma_{\beta}$ and for any
$(\varepsilon,q,p)$ close to $(\beta,q,\eta_{\beta}(q))$ we have

$\pi_{q}\circ F_{\varepsilon}(q,p)=q+m\qquad\Leftrightarrow\qquad
p=\eta_{\varepsilon}(q).$

The latter implies – together with the continuity of $\gamma$ – that for
$\varepsilon<\beta$ and sufficiently close to $\beta$, we have
$\eta_{\varepsilon}=\gamma_{\varepsilon}$.

Now consider, the map
$\Delta_{2}:(\beta-r,\beta+r)\times\mathbb{R}\to\mathbb{R}$ defined by

$(\varepsilon,q)\mapsto\pi_{p}\circ
F^{n}_{\varepsilon}(q,\eta_{\varepsilon}(q))-\eta_{\varepsilon}(q).$

Since $\eta_{\varepsilon}=\gamma_{\varepsilon}$ for
$\beta-r<\varepsilon\leq\beta$ and $\text{graph}\,\gamma_{\varepsilon}$ is
$(m,n)$-periodic, $\Delta_{2}$ vanishes on $(\beta-r,\beta]\times\mathbb{R}$.
Since $\Delta_{2}$ is analytic and $(\beta-r,\beta+r)\times\mathbb{R}$ is
connected, $\Delta_{2}$ vanishes on $(\beta-r,\beta+r)\times\mathbb{R}$.

Combining the two results on $\Delta_{1}$ and $\Delta_{2}$, we obtain that,
for any $\varepsilon\in(\beta-r,\beta+r)$, $\text{graph}(\eta_{\varepsilon})$
is an $(m,n)$-periodic graph of $F_{\varepsilon}$. This contradicts the fact
that $\varepsilon$ cannot be bigger than $\beta$. ∎

###### Proof of Theorem 8.

Let us suppose that $\mathcal{I}_{(m,n)}(\mathbb{R})$ has an accumulation
point $\beta\in I$ and show that in this case
$\mathcal{I}_{(m,n)}(\mathbb{R})=I$. The proof of Lemma 27 can be adapted to
this context: there is a sequence $(\varepsilon_{n})_{n}$ converging to
$\beta$ with $\varepsilon_{n}\neq\beta$ satisfying

$\forall(n,q)\in\mathbb{N}\times\mathbb{R}\qquad\qquad\Delta_{2}(\varepsilon_{n},q)=0.$

Hence $\Delta_{2}$ is flat at any $(\beta,q)\in\\{\beta\\}\times\mathbb{R}$,
meaning that its partial derivatives of any order in $\varepsilon$ and $q$
vanish. By analyticity of $\Delta_{2}$ and $1$-periodicity in $q$, it vanishes
on an open set of the form $J\times\mathbb{R}$. Hence for $\varepsilon$
sufficiently close to $\beta$, $\eta_{\varepsilon}$ parametrizes an
$(m,n)$-periodic $F_{\varepsilon}$-invariant graph. We conclude that
$\mathcal{I}_{(m,n)}(\mathbb{R})$ has non-empty interior, and by Lemma 27 that
$\mathcal{I}_{(m,n)}(\mathbb{R})=I$. ∎

##  6\. Proof of Theorem 15

Let $(\Omega_{\varepsilon})_{\varepsilon\in I}$ be an analytic family of
strictly convex domains with analytic boundary. By applying homotheties to the
different $\Omega_{\varepsilon}$, we can suppose that each
$\Omega_{\varepsilon}$ has perimeter $1$ (note that homotheties do not break
the property of the billiard map in a domain of having an $(m,n)$-periodic
invariant graph of a fixed rotation number).

For each $\varepsilon\in I$, one can consider a parametrization
$s\in\mathbb{R}\mapsto\gamma_{\varepsilon}(s)$ of $\gamma_{\varepsilon}$ by
arc-length which by assumption one can assume analytic in $(\varepsilon,s)$.
The billiard map inside $\Omega_{\varepsilon}$ induces an exact symplectic
twist map $F_{\varepsilon}:\mathbb{R}\times(-1,1)\to\mathbb{R}\times(-1,1)$
defined for all $(s,\sigma)\in\mathbb{R}\times(-1,1)$ by

$F_{\varepsilon}(s,\sigma)=(s_{1},\sigma_{1})$

where, if $\sigma=-\cos\varphi$ and $\varphi\in(0,\pi)$, then $s_{1}$ and
$\sigma_{1}$ are defined by following requirements:

  * (a)

$\gamma_{\varepsilon}(s_{1})$ is the second point of intersection with
$\partial\Omega_{\varepsilon}$ of the line $\ell$ passing through
$\gamma_{\varepsilon}(s)$ and making an angle $\varphi$ with the tangent line
of $\partial\Omega$ at $\gamma_{\varepsilon}(s)$;

  * (b)

the line $\ell$ makes an angle $\varphi_{1}$ with the tangent line of
$\partial\Omega$ at $\gamma_{\varepsilon}(s_{1})$ and this defines
$\sigma_{1}=-\cos\varphi_{1}$.

Let us check that the corresponding family of billiard maps
$(F_{\varepsilon})_{\varepsilon\in I}$ satisfies the assumptions of Theorem 8.

It is well-known that the billiard map written in the coordinates $(s,\sigma)$
is an exact symplectic twist map as defined in Definition 1.

Given $\varepsilon\in I$ and $(s,\varphi)\in\mathbb{R}\times(-1,1)$, the
corresponding line $\ell$ (as in (a)) is transverse to the boundary of
$\Omega_{\varepsilon}$ at $\gamma_{\varepsilon}(s_{1})$. Applying the implicit
function theorem, one deduces that there exist a neighborhood $U$ of
$(\varepsilon,s,\varphi)\in I\times\mathbb{R}\times(-1,1)$ and an analytic map
$\varphi:U\to\mathbb{R}$ such that
$s_{1}(\varepsilon^{\prime},s^{\prime},\sigma^{\prime})=\varphi(\varepsilon^{\prime},s^{\prime},\sigma^{\prime})$
for $(\varepsilon^{\prime},s^{\prime},\sigma^{\prime})\in U$. The analytic
regularity of $\sigma_{1}$ comes from the fact that it can be written as

$\sigma_{1}=\frac{\gamma_{\varepsilon}(s_{1})-\gamma_{\varepsilon}(s)}{\|\gamma_{\varepsilon}(s_{1})-\gamma_{\varepsilon}(s)\|}\cdot\gamma_{\varepsilon}^{\prime}(s_{1})$

where $u\cot v$ denotes the scalar product of two vectors $u$ and $v$. Hence
Assumption (i) of Theorem 8 is satisfied.

It is well-known that since each $\Omega_{\varepsilon}$ is strictly convex,
the map $F_{\varepsilon}$ extends to a continuous map $\mathbb{R}\times[-1,1]$
satisfying for any $s\in\mathbb{R}$ the equalities
$F_{\varepsilon}(s,0)=(s,0)$ and $F_{\varepsilon}(s,1)=(s+1,1)$. This implies
that $\text{TI}(F_{\varepsilon})=(0,1)$.

Hence the assumptions of Theorem 8 are satisfied and it implies the result.

##  7\. Proof of Theorem 19

Let $(\Omega_{\varepsilon})_{\varepsilon\in I}$ be an analytic family of
strictly convex domains with analytic boundary. We introduce the so-called
envelope coordinates on each $\partial\Omega_{\varepsilon}$, see [9]. They are
defined as follows.

By applying translations to the domains, one can assume that there is a point
$O$ which remains inside all domains. Consider a fixed direction $Ox$. For
each $\varepsilon\in I$ and any angle $\theta\in\mathbb{R}$, one can associate
the oriented line $\mathcal{L}_{\theta}$ to $\partial\Omega$, which makes an
angle $\theta+\pi/2$ with $Ox$ and is tangent to it at a point
$\alpha(\theta)$ where the orientations of $\partial\Omega$ and
$\mathcal{L}_{\theta}$ are the same. Let $p_{\varepsilon}(\theta)$ be the
distance from $O$ to $\mathcal{L}_{\theta}$. Under the assumptions of Theorem
19, one can assume that $p$ is analytic in $(\varepsilon,\theta)$.

For any point $p\in\mathbb{R}^{2}\setminus\Omega$, one can find a unique
$\theta$ such that the vector $\alpha(\theta)-z$ and $\mathcal{L}_{\theta}$
are collinear and with the same orientation. The pair $(\theta,\gamma)$ where
$\gamma=\|\alpha(\theta)-z\|^{2}/2$ is called the envelope coordinate of $p$
and uniquely determines $p$.

The outer billiard map outside $\Omega_{\varepsilon}$ acts on the space of
enveloppe coordinates $\mathbb{R}\times(0,+\infty)$ and induces an exact
symplectic twist map
$F_{\varepsilon}:\mathbb{R}\times(0,+\infty)\to\mathbb{R}\times(0,+\infty)$
defined for all $(\theta,\gamma)\in\mathbb{R}\times(0,+\infty)$ by

$F_{\varepsilon}(\theta,\gamma)=(\theta_{1},\gamma_{1}).$

Let us check that the corresponding family of outer billiard maps
$(F_{\varepsilon})_{\varepsilon\in I}$ satisfies the assumptions of Theorem 8.

Given $\varepsilon\in I$ and
$p\in\mathbb{R}^{2}\setminus\overline{\Omega}_{\varepsilon}$, consider
$G_{\varepsilon}(p)$ to be the image of $p$ after one outer billiard
reflection on $\Omega_{\varepsilon}$. The map $G$ is well-defined and
analytic. It is then a consequence of the implicit function theorem that the
enveloppe coordinates of a point $q$ depend analytically on $q$. Hence
Assumption (i) of Theorem 8 is satisfied.

It is well-known that the twist interval of a dual billiard map is
$\text{TI}(F_{\varepsilon})=(0,1/2)$. Hence the assumptions of Theorem 8 are
satisfied and it implies the result.

##  8\. Proof of Theorem 20

Let us first recall some basic properties of symplectic billiards; we refer to
[1] for more details. Consider a strictly convex bounded domain $\Omega$ with
smooth boundary and parametrized by a periodic map
$\gamma:\mathbb{R}\to\mathbb{R}^{2}$; without loss of generality, let us
assume here that the period is $1$. Given $t\in[0,1)$, there is a unique
$t^{\ast}\in[0,1)$ different from $t$ and such that the tangent lines to
$\partial\Omega$ at $\gamma(t)$ and at $\gamma(t^{\ast})$ are parallel.

The symplectic billiard map is given by a map

$\displaystyle F:\mathcal{P}$ $\displaystyle\to$ $\displaystyle\mathcal{P}$
$\displaystyle(t_{1},t_{2})$ $\displaystyle\mapsto$
$\displaystyle(t_{2},t_{3})$

where $F(t_{1},t_{2})=(t_{2},t_{3})$ if and only if $\gamma(t_{1})$,
$\gamma(t_{2})$, and $\gamma(t_{3})$ are successive reflection points and
$\mathcal{P}$ is the set

$\mathcal{P}=\\{(t_{1},t_{2})\in\mathbb{R}^{2}\,|\,t_{1}<t_{2}<t_{1}^{\ast}\\}.$

Let $\omega(x,y)=dx\wedge dy$ the canonical symplectic form on
$\mathbb{R}^{2}_{(x,y)}$. The map $S:\mathcal{P}\to\mathbb{R}$ defined by

$S(t_{1},t_{2})=-\omega(\gamma(t_{1}),\gamma(t_{2}))$

is a generating map for $S$ in the following sense: for any
$(t_{1},t_{2})\in\mathcal{P}$ and $(t_{2},t_{3})\in\mathcal{P}$,

(4)
$F(t_{1},t_{2})=(t_{2},t_{3})\quad\Longleftrightarrow\quad\partial_{2}S(t_{1},t_{2})+\partial_{1}S(t_{2},t_{3})=0.$

Let us relate the map $F$, the space $\mathcal{P}$ and the map $S$ to our
Definition 1. Observe that the map $\varphi$ defined by
$\varphi(t_{1},t_{2})=(t_{1},s_{1})$ where

$s_{1}=-\partial_{1}S(t_{1},t_{2})=\omega(\gamma^{\prime}(t_{1}),\gamma(t_{2}))$

is a diffeomorphism from $\mathcal{P}$ on its image $\varphi(\mathcal{P})$,
which is simply $\mathbb{A}_{s^{\pm}}$, with

$s^{-}(t)=\omega(\gamma^{\prime}(t),\gamma(t))\qquad\text{and}\qquad
s^{+}(t)=\omega(\gamma^{\prime}(t),\gamma(t^{\ast})).$

Therefore $F$ can be rewritten in coordinates $(t,s)$ as a map of the space
$\mathbb{A}_{s^{\pm}}$. If $F(t_{1},s_{1})=(t_{2},s_{2})$, then

$\frac{\partial t_{2}}{\partial
s_{1}}=\frac{1}{\omega(\gamma^{\prime}(t_{1}),\gamma^{\prime}(t_{2}))}>0,$

hence the twist condition is satisfied. Finally $S$ is a generating map in the
sense of Definition 1 since

$s_{2}dt_{2}-s_{1}dt_{1}=d(\omega(\gamma(t_{1}),\gamma(t_{2})))=dS,$

as it follows from (4). We deduce the following:

######  Proposition 28.

The symplectic billiard dynamics can be modelled by the exact-symplectic
twist-map $F:\mathbb{A}_{s^{\pm}}\to\mathbb{A}_{s^{\pm}}$.

Let $(\Omega_{\varepsilon})_{\varepsilon\in I}$ be an analytic family of
strictly convex domains with analytic boundary. By applying homotheties to the
different $\Omega_{\varepsilon}$, we can suppose that each
$\Omega_{\varepsilon}$ has perimeter $1$ (note that homotheties do not break
the property of the symplectic billiard map in a domain of having an
$(m,n)$-periodic invariant graph of a fixed rotation number).

Let us denote by $(\varepsilon,t)\in
I\times\mathbb{R}\mapsto\gamma_{\varepsilon}(t)$ an analytic map which is
$1$-periodic in $t$ and such that for any $\varepsilon$,
$\gamma_{\varepsilon}$ parametrizes $\partial\Omega_{\varepsilon}$. Following
Proposition 28, this induces a family of maps
$s_{\varepsilon}^{\pm}:\mathbb{R}\to\mathbb{R}$ and of exact-symplectic twist
maps
$F_{\varepsilon}:\mathbb{A}_{s^{\pm}_{\varepsilon}}\to\mathbb{A}_{s^{\pm}_{\varepsilon}}$.

It can be checked using the implicit function theorem that the family
$(F_{\varepsilon})_{\varepsilon\in I}$ is analytic and defined on
$\mathbb{A}_{I,s^{\pm}}$. Moreover the twist interval of the symplectic
billiard map is $(0,1)$ by the same continuity argument as for the classical
billiard (see Section 6).

Hence the assumptions of Theorem 8 are satisfied and this implies the result.

##  9\. Proofs of Main Theorem 2 (Theorems 12) and Theorem 17

Let us recall that $\mathscr{T}^{\omega}$ is the set of analytic twist-maps of
$\mathbb{A}_{p^{\pm}}$ which admit a continuous extension to the closure
$\overline{\mathbb{A}_{p^{\pm}}}$; the set $\mathscr{T}_{m,n}^{\omega}$
consists of analytic twist-maps of $\mathscr{T}^{\omega}$ having an
$(m,n)$-periodic invariant graph. For any $F\in\mathscr{T}_{m,n}^{\omega}$
there is an analytic $1$-periodic map $\gamma_{F}:\mathbb{R}\to\mathbb{R}$
such that $\text{graph}(\gamma_{F})$ is the unique $(m,n)$-periodic invariant
graph of the map $F$.

######  Lemma 29.

The set $\mathscr{T}_{m,n}^{\omega}$ is closed in $\mathscr{T}^{\omega}$.

###### Proof.

Let $(F_{k})_{k}$ be a sequence of maps in $\mathscr{T}_{m,n}^{\omega}$
converging to a map $F\in\mathscr{T}^{\omega}$. As a consequence of Lemma 25,
one can find an open neighborhood $U$ of
$\text{graph}\,(p^{-})\cup\text{graph}\,(p^{+})$ in $\mathbb{A}_{p^{\pm}}$
such that for any map $G$ close to $F$ and for any pair
$(q,p)\in\mathbb{A}_{p^{\pm}}$ satisfying $G^{n}_{\Omega}(q,p)=(q+m,p)$ has
the property

$(q,p)\in\mathbb{A}_{p^{\pm}}\setminus U.$

This fact together with Remark 4 imply that for $k$ large enough the graph of
the map $\gamma_{F_{k}}$ lies in the set $\mathbb{A}_{p^{\pm}}\setminus U$;
hence that the maps $\gamma_{F_{k}}$ are Lipschitz-continous with the same
Lipshitz constant. Ascoli-Arzelà theorem applies: there is a subsequence
$(\gamma_{F_{n_{k}}})_{k}$ converging to a continuous $1$-periodic map
$\gamma_{\infty}:\mathbb{R}\to\mathbb{R}$ whose graph also lies in
$\mathbb{A}_{p^{\pm}}\setminus U$ and satisfies the following identity

$\forall q\in\mathbb{R}\qquad
F^{n}(q,\gamma_{\infty}(q))=(q+m,\gamma_{\infty}(q)).$

Proposition 21 implies then that $F$ has an $(m,n)$-periodic invariant graph,
hence $F\in\mathscr{T}_{m,n}^{\omega}$. ∎

######  Lemma 30.

The map
$(F,q)\in\mathscr{T}_{m,n}^{\omega}\times\mathbb{R}\mapsto\gamma_{F}(q)\in\mathbb{R}$
is continuous for the induced topology.

###### Proof.

Let $(F_{k})_{k}$ be a sequence of maps in $\mathscr{T}_{m,n}^{\omega}$
converging to a map $F\in\mathscr{T}^{\omega}_{m,n}$. Using the same notations
and arguments as in the proof of Lemma 29, for any $k$ the map
$\gamma_{F_{k}}$ is Lipschitz-continuous and we can choose the Lipschitz-
constant uniformly in $k$. Hence it has at least one converging subsequence
for the uniform topology. Now any converging subsequence $\gamma_{\infty}$
parametrizes an $(m,n)$-periodic invariant graph of $F$ by continuity. Hence
by Proposition 21, $\gamma_{\infty}=\gamma_{F}$ and all converging
subsequences of $(\gamma_{F_{k}})_{k}$ have the same limit. This implies that
the sequence $(\gamma_{F_{k}})_{k}$ converges to $\gamma_{F}$, which concludes
the proof. ∎

###### Proof of Main Theorem 2 (Theorem 12).

As stated by Lemma 29, the set $\mathscr{T}_{m,n}^{\omega}$ is closed. Let us
prove that $\mathscr{T}_{m,n}^{\omega}$ is locally given by the zeros of an
analytic map. Let $F_{0}\in\mathscr{T}_{m,n}^{\omega}$ and consider its
$(m,n)$-periodic invariant graph $\gamma_{F_{0}}$. Since the latter has no
conjugate points, by the implicit function theorem on analytic maps of Banach
spaces (see [37]) there exists a neighborhood $V$ of $F_{0}$ in
$\mathscr{T}^{\omega}$ and an analytic map

$\overline{p}:V\times\mathbb{R}\to\mathbb{R}$

such that

  1. (1)

for any $F\in V$, $q\mapsto\overline{p}_{\Omega}(q)$ is $1$-periodic;

  2. (2)

$\overline{p}_{F_{0}}=\gamma_{F_{0}}$;

  3. (3)

for any map $F$ in $V$ close to $F_{0}$, and any point $(q,p)$ close to
$(q,\gamma_{F_{0}}(p))$, we have

$\pi_{1}\circ F^{n}(q,p)=q+m\qquad\Leftrightarrow\qquad
p=\overline{p}_{F}(q).$

By Lemma 30, its means that we can eventually shrink $V$ and state that
$\overline{p}_{F}=\gamma_{F}$ for $F\in\mathscr{T}_{m,n}^{\omega}\cap V$.
Hence we can state the equivalence

$\forall F\in V\qquad
F\in\mathscr{T}_{m,n}^{\omega}\quad\Longleftrightarrow\quad\forall
q\in\mathbb{R}\quad\pi_{2}\circ
F^{n}(q,\overline{p}_{F}(q))=\overline{p}_{F}(q).$

Thus consider the space Banach space $\mathscr{C}(\mathbb{R}/\mathbb{Z})$ of
continuous $1$-periodic maps on $\mathbb{R}$, and define the map

$\varphi:\left\\{\begin{array}[]{rcl}V&\to&\mathscr{C}(\mathbb{R}/\mathbb{Z})\\\
F&\mapsto&(q\mapsto\pi_{2}\circ
F^{n}(q,\overline{p}_{F}(q))-\overline{p}_{F}(q))\end{array}\right.$

The map $\varphi$ is analytic by construction and satisfies
$\varphi^{-1}(0)=\mathscr{T}_{m,n}^{\omega}\cap V$. Since this is true for any
$F_{0}\in\mathscr{T}_{m,n}^{\omega}$, this concludes the proof. ∎

###### Proof of Theorem 17.

Given a domain $\Omega\in\mathscr{D}_{r}$, we can consider the billiard map
$F_{\Omega}$ inside $\Omega$. As a consequence of the implicit function
theorem for analytic maps of Banach spaces (see [37]), the map
$F:\mathcal{D}_{r}\to\mathscr{T}^{\omega}$ is an analytic. Moreover, the set
of domains in $\mathscr{D}_{r}$ having an $(m,n)$-periodic invariant graph is
$X_{m,n}:=F^{-1}(\mathscr{T}^{\omega}_{m,n})$. Since
$\mathscr{T}^{\omega}_{m,n}$ is an analytic subset of $\mathscr{T}^{\omega}$,
the set $X_{m,n}$ is an analytic subset of $\mathscr{D}_{r}$. ∎

##  Appendix A Remarks on billiard dynamics

In this section we underline the peculiarity of billiard maps by proving
properties on their $(m,n)$-periodic graphs, as given in Proposition 21 for
general twist-maps, but in a different way which requires less machinery. It
relies on the fact that the generating map $h$ of a billiard is the opposite
of the distance between consecutive points, and hence if one fixes two points
$x$ and $z$, the quantity

$h(x,y)+h(y,z)$

is minimized for $y\neq x,z$.

######  Proposition 31.

Let $f:\mathbb{T}^{1}\times(0,1)\to\mathbb{T}^{1}\times(0,1)$ be the billiard
map written in $(s,-\cos\varphi)$ coordinates and $\Gamma$ be an
$(m,n)$-periodic Lipschitz graph $\Gamma$ of $f$. Then:

  * (i)

$\Gamma$ invariant by $f$;

  * (ii)

the projection of an orbit of $\Gamma$ is a minimal configuration;

  * (iii)

$\Gamma$ is as smooth as $f$ is.

###### Proof.

Items (i) and (iii) follow from Proposition 21.
Let us show item (ii) First let us show that the projection
$\underline{s}=(s_{p})_{p\in\mathbb{Z}}$ of an orbit
$(s_{p},y_{p})_{p\in\mathbb{Z}}$ and $(s_{0},y_{0})\in\Gamma$ is minimal (here
$y$ stands for $-\cos\varphi$).

Let us write $\Gamma=\\{(s,\phi(s))\,|\,s\in\mathbb{R}/L\mathbb{Z}\\}$ where
$\phi:\mathbb{T}^{1}\to(0,1)$ is a Lipschitz continuous map and $L$ is the
perimeter of the billiard boundary. Denote by $h:D\to\mathbb{R}$ the
generating function of the billiard where $D=\\{(s,s^{\prime})\,|\,s\leq
s^{\prime}\leq s+1\\}$. For any $p\leq q$, define

$\mathcal{E}_{pq}(x_{p},\ldots,x_{q})=\sum_{k=p}^{q-1}h(x_{k},x_{k+1})-\int_{x_{p}}^{x_{q}}\phi(u)du.$

In the case of the billiard map, $\mathcal{E}_{pq}$ is well-defined and
continuous on the compact set
$K=\\{(x_{p},\ldots,x_{q})\,|\,x_{0}\in[0,1],\,\forall k\,\,x_{k}\leq
x_{k+1}\leq x_{k}+1\\}.$ Hence it has a global minimal value which is reached
at a certain point $\underline{x}=(x_{p},\ldots,x_{q})$. Now by triangular
inequality, we have the important fact that

$\underline{x}\in\text{int}(K).$

This property is specific to the billiard map and due to the fact that
$h(x,y)+h(y,z)<h(x,x)+h(x,z)$ for any $x<y<z$. Hence $\underline{x}$ is a
critical point of $\mathcal{E}_{pq}$ and by a classical argument,
$\underline{x}$ is a stationary configuration corresponding to an orbit
$(x_{j},y_{j})_{p\leq j\leq q}$ of the billiard map such that
$y_{p}=\phi(x_{p})$ and $y_{q}=\phi(x_{q})$.

It follows that our initial configuration $\underline{s}$ minimizes each
$\mathcal{E}_{pn,qn}$ among all configurations with the same endpoints
$s_{pn}$ and $s_{qn}$, where $p\leq q$. Hence $\underline{s}$ minimizes the
action considered between the indices $pn$ and $qn$, among all configurations
with the same endpoints. By another classical argument, $\underline{s}$ is
minimal. ∎

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