Chaotic Motion in the Breathing Circle Billiard

We consider the free motion of a point particle inside a circular billiard with periodically moving boundary, with the assumption that the collisions of the particle with the boundary are elastic so that the energy of the particle is not preserved. It is known that if the motion of the boundary is regular enough then the energy is bounded due to the existence of invariant curves in the phase space. We show that it is nevertheless possible that the motion of the particle is chaotic, also under regularity assumptions for the moving boundary. More precisely, we show that there exists a class of functions describing the motion of the boundary for which the billiard map has positive topological entropy. The proof relies on variational techniques based on the Aubry–Mather theory.


Introduction
A mathematical billiard with moving boundary is a region of the plane instantaneously bounded by a closed curve changing with time. The billiard problem then consists of the free motion of a point particle inside this region colliding elastically with the moving boundary.
The study of time-dependent billiards can be motivated physically by the study of confined Lorentz gas or by some models in nuclear physics (see, for instance, [5,6,13,31]). In the physical contexts, a relevant question is whether the elastic bounces can make the energy of the particle grow infinitely. The question was raised by Fermi [10] trying to explain the high velocity gained by photons. A mathematical formulation of the problem was proposed by Ulam and is now called the Fermi-Ulam model. It describes the free motion of a particle between two parallel walls moving periodically. "Fermi acceleration" then occurs if the elastic bounces with the moving walls make the energy of the particle tend to infinity. It is known that it depends on the regularity of the motion of the walls. Actually, it was proved in [17] that if the motion of the walls is at least C 6 then KAM theory applies and the energy remains bounded. On the other hand, in [33] it is shown how to construct a motion of the walls that is only continuous and allows the energy to grow to infinity (see also [21] for a similar result in a related impact model).
Time-dependent billiards can be seen as natural generalisations of the Fermi-Ulam model, and the question on the existence of Fermi acceleration naturally extends to this context. In this case, the answer also depends on the geometry of the boundary.
If the boundary is a moving ellipse, then it has been proved in [9] that it is possible to construct orbits that gain energy. Moreover, the existence of unbounded motions is a symptom of complex dynamics. In fact, in [9] it is also proved that the phenomenon of splitting of the separatrices occurs and a scattering map can be defined. We refer to [12] for more insight on the topic of Fermi acceleration in general time-dependent billiards.
Instead, if the boundary is a circle with radius varying smoothly with time, then KAM theory applies and the energy remains bounded [29]. Since the motion along a diameter is described by the classical Fermi-Ulam model, the counterexample constructed in [33] shows that the regularity of the motion of the boundary is a fundamental assumption in [29].
In this paper, we deal with the case of a region with circular boundary of radius R(t) periodic in time. The region is called the breathing circle billiard. We show that chaotic phenomena can occur also if the motion of the boundary is regular. More precisely, we find a class of function R such that if R ∈ R then the associated billiard map has positive topological entropy. The class R has a somehow technical definition, but it can be shown that a representative is with M sufficiently large with respect to δ.
The dynamics of a time-dependent billiard whose boundary remains a convex curve can be described by a 4-dimensional exact symplectic map [15]. However, for the breathing circle billiard, the angular momentum is a first integral so that the dynamics can be reduced to a two-dimensional map of the cylinder that turns out to be exact symplectic and twist at large energies. We show that to the reduced map we can apply the variational framework of Aubry-Mather theory, which implies the existence of interesting invariant sets. In particular, a key role is played by invariant curves with irrational rotation numbers. More precisely, it is known that the lack of invariant curves for a given irrational rotation number implies the existence of chaotic motion, and in particular we apply a result by Forni [11] to obtain the existence of many invariant measures with positive metric entropy (and support of zero Lebesgue measure), from which positiveness of topological entropy follows. See [1,2] for related results.
In the last decades, many results have been proved in the direction of "breaking" invariant curves, also in higher dimensions, giving rise to the so-called converse KAM theory [14,18,19]. In this paper, we show the nonexistence of some invariant curve using a criterion based on the variational approach of Aubry-Mather theory, in the spirit of what is done in [24]. More precisely, it is known that orbits of exact symplectic twist maps correspond to stationary points of an action functional, and the ones on invariant curves are minimal. As a consequence, the second variation of the action must be positive on orbits lying on invariant curves. The main idea to prove our main result is then to show that if R ∈ R the second variation of the action is negative in a zone of the phase space, preventing the existence of invariant curves for some irrational rotation numbers. From a technical point of view, in order to compute the second variation of the action, one needs the generating function of the associated diffeomorphism. A considerable part of the paper is dedicated to get an explicit formulation of the generating function of the billiard map for large energies. To this aim, we follow the idea, used in [16] for the (non-periodic) Fermi-Ulam model, that the generating function is given by the Lagrangian action of a solution of the Dirichlet problem between two consecutive impacts. We conclude noting that a consequence of our approach is the existence of Aubry-Mather sets with different rotation numbers, giving rise to periodic and quasi-periodic motions of the breathing circle billiard. Similar results have been obtained for other systems with impacts such as bouncing balls [20].
The paper is organised as follows: In Sect. 2, we state the problem and the main results of the paper. In Sect. 3, we study the Dirichlet problem between two consecutive impacts, and its results are used in Sect. 4 to compute the generating function of the billiard map. In Sect. 5, we describe the periodic and quasi-periodic motions. Section 6 is dedicated to the proof of the main theorem on chaotic motion. Appendix A contains the proof that the class of functions R is not empty. Finally, the main results of Aubry-Mather theory used in the paper are collected in Appendix B.

Statement of the Problem and Main Results
Let R(t) be a strictly positive function, and let the breathing circle D t be the bounded region of the plane with moving boundary ∂D t = {x ∈ R 2 : |x| 2 = R 2 (t)}. Let us consider a particle of unitary mass moving freely inside D t and satisfying the elastic impact law at every bounce on the boundary. Assume that ∂D t is positively oriented and denote byτ ,η the unitary tangent and outward normal vectors at points of ∂D t . By a bouncing solution, we mean a continuous and piecewise-C 2 function x : R → R 2 with a sequence of impact times (t n ) n∈Z satisfying 1.ẍ(t) = 0 for t ∈ (t n , t n+1 ) for every n ∈ Z, 2. |x(t)| < R(t) for t ∈ (t n , t n+1 ) for every n ∈ Z, 3. |x(t n )| = R(t n ) for every n ∈ Z, 4.ẋ(t + n ) ·τ n =ẋ(t − n ) ·τ n andẋ(t + n ) ·η n = −ẋ(t − n ) ·η n + 2Ṙ(t n ), whereτ n ,η n are the unitary tangent and outward normal vectors at x(t n ), andẋ(t − n ) andẋ(t + n ) denote the velocity vector just before and after the bounce at time t n , respectively. Condition 4 describes the elastic bouncing condition: the tangent component of the velocity is preserved and the impulse is given in the normal direction. Note that ifṘ ≡ 0 we get the usual mirror law.
To describe the possible bouncing solutions, we first note that the angular momentum is preserved, more precisely, Proposition 2.1. Let x(t) be a bouncing solution with impact times (t n ) n∈Z , and then, the angular momentum Proof. It is clear that the angular momentum is constant for t ∈ (t n , t n+1 ) for every n ∈ Z. Moreover at the bouncing time it holds Since the motion is in the plane we have C(t) = (0, 0, c). If c = 0, the motion is along a diameter and never leaves the diameter. Moreover, there is a symmetry between motion in the clockwise and in the anticlockwise direction given by changing sign to c (see (3.1)). Without loss of generality in the following we assume c > 0, which corresponds to anticlockwise motion.
Fixed the angular momentum c and assuming that the function R(t) is periodic, a bouncing solution can be obtained as the orbit of a map, the breathing circle billiard map, on a two-dimensional cylinder. Since R(t) is not constant, the radial velocity of the particle has varying moduli so that a good choice of coordinates is given by the time of an impact and the radial velocity after an impact. Other velocity-related coordinates for the second component work as well. If R ∈ C k with k > 7, it has been shown in [29] that the billiard map admits invariant circles which are homotopically non-trivial, thus showing that the velocity of a particle remains bounded. Their proof follows by showing that for high velocities the billiard map can be written as a small perturbation of an integrable map. Our main result in the case of highly regular functions R may be read as a completion of that in [29], as we prove the coexistence of regular and of chaotic orbits in the phase space. However, our results hold under more general assumptions. If R ∈ C 2 , we prove existence of both chaotic and regular motions. An informal statement is the following.
The formal statements along with the definitions of the class R and of quasi-periodic bouncing solutions are given in Sect. 2.1. We now discuss the geometric ideas underlying the proofs of this result.
Let us first consider the simple case in which R(t) ≡ R * > 0, that is the circle D is fixed. It is well known that in this case the norm of the velocity v(t) := |ẋ(t)| of a solution and the angle ϕ of reflection at ∂D are preserved at every bounce. Hence, the radial velocity of the particle is constant and a good set of coordinates for the billiard map is given by the arc length s of ∂D and the angle ϕ. The motion is then described by the billiard map B(s 0 , ϕ 0 ) = (s 1 , ϕ 1 ) of the cylinder R/2πR * Z × (0, π 2 ) given by The corresponding orbits (s 0 +2R * nϕ 0 , ϕ 0 ) n∈Z represent the successive impact points on the boundary and the respective angles of reflection. It is clear that a trajectory is uniquely determined (up to rotations) by the value ϕ 0 , and introducing the rotation number ω = πϕ 0 ∈ (0, 1 2 ) one finds that if ω = p/q ∈ Q then the trajectory closes after q bounces and p revolutions, while if ω ∈ R \ Q then the trajectory never closes and the bouncing points are dense on the boundary.
In addition, the map B is exact symplectic in the sense that for all This implies that it is implicitly given by means of a generating function h : Finally, one observes that the (positive) twist condition is equivalent to for all (s 0 , s 1 ) ∈ R 2 , which implies that (2.1) and (2.2) are equivalent. Exact symplectic (positive or negative) twist maps enjoy several properties that can be deduced only by the properties of the generating function and are at the basis of Aubry-Mather theory (see Appendix B).
In the first part of this paper, we follow the approach in [16] to find an exact symplectic twist map that describes the motion of some bouncing solutions of the breathing circle billiard. However, it turns out that the variables (s, ϕ) are not a good choice to this aim, but as in [29] it is better to work with time and velocity. Again in the case of the fixed circle D with R(t) ≡ R * , consider the variables (t, v 2 (t)) where t represents the bouncing time. By Proposition 2.1, fixed the angular momentum c > 0, we can consider the map B c (t 0 , v 2 0 ) = (t 1 , v 2 1 ) of the cylinder R/Z × R + , depending on the parameter c, given by where v 2 i := v 2 (t i ). Here, f c (v 2 ) plays the role of the rotation number. Notice that the map B c is still exact symplectic, since v 2 where F c is a primitive of f c . However, it does not satisfy the twist condition, as f c (v 2 ) has one change of sign. This has the following geometric interpretation: fixing c there exist two possible trajectories joining two points on the boundary in time t 1 − t 0 . More precisely, hence using c = R * v cos ϕ one gets two values for the angles ϕ ± of reflection. This problem can be overcome by considering only motions with large values of v 2 . Actually, it can be verified that for v 2 large, B c (t 0 , v 2 0 ) satisfies the (negative) twist condition and can be expressed implicitly as in (2.2) by using the generating function Thus, one obtains an exact symplectic twist map, which describes the motion of some bouncing solutions of the fixed circle billiard.
The main idea behind the present paper is to replicate this last construction in the case of the breathing circle D t . Given a C 2 , strictly positive and 1-periodic function R(t) and fixed the angular momentum c, we show that there exist variables (t, K) in the cylinder R/Z × R + such that the billiard map P (t, K) is exact symplectic and twist for K large enough. The variable K is a generalisation of v 2 and reduces to v 2 2 for R(t) constant. This program is developed in Sects. 3 and 4 where the generating function is obtained by using the solutions of a suitable Dirichlet problem.
Thus, we show that Aubry-Mather theory can be applied to the exact symplectic twist map P , and the last part of the paper is devoted to proving the main result.
First of all, one gets the existence of interesting invariant sets, called Aubry-Mather sets, whose properties are summarised in Appendix B. Aubry-Mather sets are characterised by their rotation number ω; hence, we denote them by M ω . If ω = p/q ∈ Q; then, M ω contains a periodic orbit, instead if ω ∈ R \ Q then M ω is a graph over the first coordinate of the cylinder R/Z × R + and is either an invariant curve which is homotopically non-trivial or a Cantor set. Orbits on M ω give rise to interesting bouncing solutions of the breathing billiards: in the case ω = p/q ∈ Q one gets periodic solutions of period p with q bounces in a period, while in the case ω ∈ R \ Q one gets families of quasi-periodic solutions of frequencies (1, 1/ω) (see Definition 2.4).
By the properties of the billiard map P , we thus obtain existence of periodic and quasi-periodic orbits for some values of the rotation number ω. In fact, since we can only guarantee that the map P is exact symplectic and twist for large values of the coordinate K, we can apply Aubry-Mather theory to P only for a subset of the possible rotation numbers and we need to assume that R belongs to the class R defined in Definition 2.2. This is discussed in Sect. 5 where we give the proof of our first result, Theorem 2.5.
Then, Aubry-Mather theory gives also the tools to prove the existence of chaotic motion for exact symplectic twist maps on a cylinder. Namely it is well known (see Theorem B.6) that if an Aubry-Mather set of irrational rotation number is a Cantor set, then close to it there exist many invariant probability measures with positive metric entropy; hence, the map has positive topological entropy. Unfortunately the supports of these measures have zero Lebesgue measure in the cylinder, so that the problem of proving chaotic behaviour in the sense of smooth ergodic theory remains open.
In order to prove that an Aubry-Mather set with irrational rotation number is a Cantor set, as explained before, one needs to prove that it cannot be an invariant curve. Many techniques are available to prove the non-existence of invariant curves and form the so-called converse KAM theory. Here we are using the variational characterisation of the invariant curves for exact symplectic twist maps. We recall that there exists an action functional for which the orbits of an exact symplectic twist map are critical points, and in particular each orbit lying on an invariant curve is a point of minimum. As a consequence, the second variation of the action along an invariant curve is positive (see the function in Proposition B.7-(ii)). The basic idea to prove that a map does not have an invariant curve in a region of the cylinder is then to show that the second variation is negative in the whole region. We apply this method to the map P as showed in Sect. 6. For us, it is enough to find a vertical segment in the cylinder on which the second variation of the action is negative, so that cannot be crossed by any invariant curve. Then we show that there exists a set I ⊂ R such that the Aubry-Mather sets of the map P with rotation number ω ∈ I must cross the vertical segment . This is achieved by recalling that the rotation number gives an estimate on the zone of the cylinder on which the corresponding Aubry-Mather set is located. In the end, we have showed that the Aubry-Mather sets M ω with ω ∈ I cannot be invariant curves, and this implies our main result stated in Theorem 2.7, namely that the map P is chaotic in the sense that it has positive topological entropy. We remark that the proof of this result needs the assumption that the angular momentum c is small enough and that the boundary of the circle D t is described by a function in R R. This last restriction is used to study the sign of the second variation of the action functional.
As a final comment, we remark that on one side the class R contains "natural" choices of behaviours of the breathing circle such as the functions R δ,M (t), which give rise to large circles with relatively small and slow breathing, thus intuitively describing a system which for large velocities is a small perturbation of the integrable system given by the billiard in a fixed circle. On the other side, we cannot say that the class R is optimal to obtain the existence of chaotic motion for the breathing circle billiard. Its technical definition mainly follows from the methods of our proofs.
It is worth noting that the unperturbed map B c is exact and the twist has exactly one change of sign, so that it is a non-twist map as defined in [7,8]. However, it is not clear if also the billiard map P is a non-twist map as we only prove that it can be expressed via a good generating function only for large values of the coordinate K.

Formal Statement of the Main Result
We split our main result into two theorems: Theorem 2.5 refers to periodic and quasi-periodic motions, while Theorem 2.7 concerns the chaotic behaviour.
Let us first introduce the classes of functions we consider for the motion of ∂D t .

Definition 2.2.
Let R(t) be a C 2 , strictly positive and 1-periodic function, and ε ∈ (0, 1) be a fixed parameter. We denote by · the sup-norm of a function and use the notations We say that R(t) belongs to the class R if σ > 2. We say that R(t) belongs to the class R if: and there existst ∈ [0, 1) such that (R(t)R + Ṙ R) < 0 and (ii) For simplicity, we drop the dependence on ε in the notations for σ and the classes R and R. Clearly R ⊂ R and the classes R and R are non-empty as shown by the next proposition whose technical proof is in Appendix A. In the following, the parameter ε is to be considered fixed in (0, 1); there is not a more interesting value for it. However, notice that the class R is empty if ε = 1 (see Remark A.2).

Proposition 2.3. The functions
) and M large enough belong to the class R. We now begin to state our main results. The first concerns the existence of regular motion for the billiard dynamics inside D t . We first give the definition of periodic and quasi-periodic bouncing solutions, inspired by [32] (see also [30]): Given ω ∈ R, we say that a family of bouncing solutions , and if ξ is a point of continuity, then so are ξ ± ω and ξ ± 1; and if ω = p/q ∈ Q then t n+q = t n + p for every n ∈ Z. If ω ∈ R \ Q and the function (ξ, a, b) → Φ ξ (a, b) is continuous, then these solutions are called classical quasi-periodic.
If ω = p/q ∈ Q, then for every ξ ∈ R so that the bouncing solution makes q bounces in time p before repeating itself. They are said (p, q)-periodic.
and let R(t) ∈ R. Then, for every 1 < ω < σ − 1 there exists a family of generalised quasi-periodic bouncing solutions of frequencies (1, 1/ω). Remark 2.6. Under the hypotheses of Theorem 2.5 we are not able to guarantee that the solutions are classical quasi-periodic. Moreover, the restriction ω > 1 is not optimal and is due to the techniques used in the proof.
Let us now consider the existence of chaotic dynamics for the billiard motion in D t . Let P be the breathing circle billiard map as defined in Proposition 4.4. Theorem 2.7. Suppose that R(t) ∈ R. Then, for c sufficiently small, the breathing circle billiard map P has positive topological entropy.
The result is proved in Sect. 6. It follows by a standard application of the Variational Principle, showing the existence of invariant probability measures with positive metric entropy. Such measures are obtained by applying results in [11]. We remark that the union of the supports of these invariant measures has zero Lebesgue measure. However, the existence of invariant probability measures with positive metric entropy and support of positive Lebesgue measure is a major problem in dynamical systems and is an open problem also for well-known twist maps of the cylinder as the standard map.
Remark 2.8. We have stated Theorems 2.5 and 2.7 for positive values of the angular momentum c, the case c = 0 corresponding to motions along a diameter and the system reducing to the Fermi-Ulam model. Our arguments can be readily adapted to the case c = 0, so that the conclusions of our main theorems also hold for the Fermi-Ulam model.

The Dirichlet Problem
The proofs of our main results rely on Aubry-Mather theory, for which we need to define a generating function for the billiard map associated with the billiard flow in D t . The first step is the study of the Dirichlet problem associated with the flow between two consecutive bounces.
Let us consider two consecutive bounces for the billiard motion in D t at times t n and t n+1 ; the Dirichlet problem which describes the billiard motion between the two bounces is from which it is evident the symmetry with respect to the change of sign of c. Bouncing condition 4 in these coordinates readṡ In the following result, we find sufficient conditions for a solution of system (3.1) to exist. Note that these solutions do not satisfy in general the bouncing conditions (3.2)-(3.3) when glued together. Proposition 3.1. Let ε ∈ (0, 1) be a fixed parameter, and let us fix a value of c > 0. For all times t n , t n+1 satisfying where E is the energy of the system, and can be integrated giving the general solution for B ∈ R. System (3.1) is rotationally invariant so that we can fix θ(t n ) = 0 without loss of generality. With this assumption, solution (3.10) represents a straight trajectory in the billiard table starting from the point (R n , 0) with velocityẋ(t n ) satisfyinġ Moreover, notice that by (3.10) (3.12) Let := A(t n + B), then the straight trajectory (3.10) is parametrised by Rn and the condition r(t n+1 ) = R n+1 is then equivalent to We then obtain which is well defined thanks to (3.4). Using (3.11), it also holds which in (3.10) gives r(t n ) = R n . We still have to verify conditions (3.7), (3.8) and Using (3.12), the first condition in (3.7) reduces tȯ From (3.13), it is immediate that the choice + has to be discarded and − is admissible. Let us now consider the second condition in (3.7). From (3.12), we need to check thaṫ But, by (3.14) we have so that both conditions (3.7) are satisfied only for − . Moreover, from (3.5), from which, using the first condition in (3.7), we geṫ that is the first condition in (3.8). The second condition is proved similarly. It remains to show that r(t) < R(t) for all t ∈ (t n , t n+1 ). From (3.8), it follows that Moreover from (3.12) and (3.14) where in the last inequality we have used (3.4). It follows that t n+1 ). This concludes the proof of the proposition, for which the solution is given by (3.10) with A, B as in (3.14) with = − as defined in (3.13).

The Generating Function
In this section, we define a generating function h(t 0 , t 1 ) for the billiard map between two consecutive bounces on ∂D t at times t 0 and t 1 . The orbits of the corresponding billiard map generate bouncing motions with large velocity, in the sense that conditions (3.7), (3.8) are satisfied. Following the approach in [16], we first show that a good guess for h is given by the action of system (3.1). Let h c (t n , t n+1 ) = tn+1 tn L c (r(t; t n , t n+1 ),ṙ(t; t n , t n+1 ))dt (4.1) where r(t; t n , t n+1 ) is the solution to (3.1) found in Proposition 3.1 and is the reduced Lagrangian of the system for a fixed value c > 0. In particular, r(t; t n , t n+1 ) satisfies the Euler-Lagrange equation For simplicity, let us denote r(t) = r(t; t n , t n+1 ) and remove the dependence on c. We have where we first used integration by parts, and then, the fact that r(t) satisfies the Euler-Lagrange equation and finally denoted ∂L ∂ṙ and analogously for ∂L ∂ṙ (t n ). Differentiating with respect to t n the relations r(t n+1 ; t n , t n+1 ) = R(t n+1 ) and r(t n ; t n , t n+1 ) = R(t n ), we get ∂r ∂t n (t n+1 ) = 0,ṙ(t + n ) + ∂r ∂t n (t n ) =Ṙ(t n ) .

Hence, we have
and, remembering the expression of L in (4.2) and that r(t n ) = R(t n ), Analogously, one can get Therefore, we conclude that if a sequence (t n ) satisfies where ∂ 1 and ∂ 2 denote differentiation with respect to the first and the second argument, respectively, then 1 2ṙ from which we get the bouncing condition (3.3) using thatṙ(t + n ) <Ṙ(t n ) anḋ r(t − n ) >Ṙ(t n ) (see (3.8)). Conversely, a sequence satisfying (3.3) also satisfies (4.4). Using also Remark 3.2, we have proved the following  We are now ready to give an explicit expression to the function h c in terms of the times of bouncing.

Proposition 4.2.
Fixed a value c > 0 for the angular momentum, let t 0 , t 1 be two consecutive bouncing times and R 0 := R(t 0 ) and R 1 := R(t 1 ) be the corresponding radii of the breathing circle. The generating function defined in (4.1) computed along solutions to (3.1) defined in (3.15) has the form Proof. Using (3.9) for solutions to (3.1) it holds dt .

Moreover by (3.15) for θ(t; t 0 , t 1 ) and Remark 3.3, we have
the proof is finished, since the generating function is defined up to an additive constant.
We now argue on the other direction. Given the function h(t 0 , t 1 ) in (4.5), we show that when restricted to a suitable subset, it is a generating function of the bouncing motion of a particle inside the breathing circle D t .

Let us consider the strip
Proof. Properties (i) and (ii) follow immediately from the regularity and the periodicity of R(t), and from the definition of h(t 0 , t 1 ).
To prove (iii), by standard computations we obtain We can then conclude by the following estimates. First, ifṘ(t 0 ) ≥ 0, then If insteadṘ(t 0 ) < 0, then we use (t 1 − t 0 ) < σ and write so that the first term in ∂ t0t1 h(t 0 , t 1 ) is positive in all cases. For the second term, we have is negative both ifṘ(t 1 ) ≤ 0, as sum of non-positive terms with R 1 > 0, and ifṘ(t 1 ) > 0 since by (t 1 − t 0 ) < σ it holds Finally, from (4.7) we find Let Ω ⊂ R 2 be the strip defined in Proposition 4.3 and rewrite h : Ω → R as where R 0 := R(t 0 ), R 1 := R(t 1 ), and c ∈ (0, ε R 2 σ ). As before let ∂ 1 and ∂ 2 denote the partial derivatives with respect to the first and second argument of a function, and let T denote the one-dimensional torus.

Proposition 4.4. Define σ * := max t∈T ∂ 1 h(t, t + σ). The equations
define implicitly a C k embedding P : T × (σ * , +∞) → T × R, P (t 0 , K 0 ) = (t 1 , K 1 ). Moreover, P is twist in the sense that ∂t 1 ∂K 0 < 0, and exact symplectic in the sense that there exists a C k function V : T×R → R such that Proof. Using that ∂ 1 h(t 0 , t 1 ) → +∞ as (t 1 − t 0 ) → 0 + as can be shown from (4.6) and Proposition 4.3-(iii), we can apply the implicit function theorem to the first of (4.8) and get the C k function for (t 0 , K 0 ) ∈ T × (σ * , +∞). Inserting (4.9) into the second of (4.8), we get the desired C k map P . To prove that it is injective, we note that if P (t 0 , K 0 ) = P (t 0 , K 0 ) then, using once again Proposition 4.3-(iii), the second of (4.8) implies t 0 = t 0 that, substituted in the first gives also K 0 = K 0 . By implicit differentiation of the first, we get the twist condition: Remark 4.5. It follows from the previous Proposition that a sequence (t n , K n ) is an orbit of the map P if and only if, for every n, (t n+1 − t n ) ∈ Ω and We finally note that the orbits of the diffeomorphism P give rise to bouncing solutions of the billiard map in the breathing circle D t . Proposition 4.6. Let (t n , K n ) be an orbit of P , let us set R n := R(t n ) and defineṙ(t + n ) asṙ . Since we obtain thatṙ(t + n ) defined in (4.10) can be written aṡ where we have used that since (t n+1 − t n ) < σ. A straightforward computation shows thatṙ(t + n ) is then equal toṙ(t + n ; t n , t n+1 ), the velocity of solution (3.15) to system (3.1) found in Proposition 3.1. In particular, (4.11) shows thatṙ(t + n ) satisfies the first condition in (3.8).
Remark 4.7. The definition ofṙ(t + n ) in Proposition 4.6 is inspired by (4.3). Notice that the two solutions of (4.3) arė which can be written aṡ Hence,ṙ − is the only solution of (4.3) which may represent the radial velocity of a bouncing solution leaving the boundary. Moreover,ṙ − = −ṙ + + 2Ṙ(t n ); hence,ṙ + andṙ − can be interpreted as the radial velocity before and after the bounce, respectively. Remark 4.8. For completeness, we show that the map P defined in Proposition 4.4 corresponds to the map M considered in [29] with a different choice of variables. The variables used in [29] are (t, I), and the map M : (t 0 , I 0 ) → (t 1 , I 1 ) is implicitly given by ⎧ ⎨

in our notations. The second equation is obtained by
and using (3.16) and (4.11) forṙ(t + 0 ). On the other hand, from Proposition 4.6 and (4.8)ṙ and, arguing as in the proof of Proposition 4.3, it holds Therefore, using . (4.12) Using (4.12), we get and again by (4.11) forṙ(t + 0 ), we conclude which is the first equation.

Periodic and Quasi-Periodic Orbits
In this section, we give the proof of Theorem 2.5. The proof is based on the Aubry-Mather theory, and the results we use are recalled in Appendix B. We begin with a preliminary result for the billiard map. Let ε ∈ (0, 1) be a fixed parameter as in Definition 2.2.
Let us now come to the Proof of Theorem 2.5. Fix c ∈ 0, ε R 2 σ . Consider ω ∈ R\Q and the corresponding functions φ, η : R → R given by Corollary 5.2. Denote by x ξ (t) = (r(t), θ(t)) ξ the bouncing solution with angular momentum c which satisfies .
and Φ ξ (t, t/ω) = x ξ (t). So that the solutions satisfy condition (i) in Definition 2.4, since the regularity follows from (5.5) and Corollary 5.2. Finally, condition (ii) follows from the definition of the rotation number of a minimal orbit.

Chaotic Motion
In this section, we prove Theorem 2.7, showing the existence of chaotic motion for the billiard map inside the breathing circle D t with R(t) ∈ R. In particular, we are going to prove that there exists c 0 > 0 such that for every c ∈ (0, c 0 ) there exist many probability measures with positive metric entropy, which are invariant for the map P defined in Proposition 4.4. As in the previous section, we will refer to results from Appendix B. The idea of the proof is the following. First we extend the map P to the whole cylinder as in the proof of Proposition 5.1. The key point is then to prove that there exists an open interval I ⊂ R such that for sufficiently small values of c, the extended map has no invariant curve with rotation number ω ∈ I. Hence, for irrational ω ∈ I the Mather sets M ω of Theorem B.4 are Cantor sets. Then, Theorem B.6 guarantees the existence of invariant probability measures with positive metric entropy for the extended map. The final step is to show that the extension has been made in such a way that these invariant measures are supported in the zone of the cylinder where the extended map coincides with P .
Let us first state and prove a series of technical lemmas. Let R ∈ R and recall by Definition 2.2 that σ > 4. Moreover, in this section we use the notationsṘt andRt forṘ(t) andR(t), respectively. Let us consider the set Proof. Let us first note that from which, by conditions (i) and (ii) of Definition 2.2, there exist 3 < ω − < ω + < σ − 1 such that every ω ∈ (ω − , ω + ) satisfies Since ω > 3, using the first of (6.1) that proves the third inequality in the definition of the set Ξ R . Analogously, since ω > 1, using the second of (6.1) that proves the first inequality in the definition of the set Ξ R . The second inequality can be easily proved.
and o(c) represents a function depending on R, ω, c, t 0 , t 1 that tends to zero uniformly for ω ∈ I as c → 0 + .
Proof. Let (t n , K n ) be an orbit of the billiard map with rotation number ω on the invariant curve Γ. From (4.8) and (4.6), a direct computation gives for the point (t 0 , K 0 ) of the orbit can be bounded by a constant depending on R and c but not on ω. Actually, from (B.2) and the fact that ω ∈ I, so that |t 1 (t 0 , K 0 , c) − t 0 | is uniformly bounded on every invariant curve with rotation number ω ∈ I for c fixed. Solving (6.2) for (t 1 − t 0 ), we get that used in (6.3) gives Since this argument applies to all points of Γ, the proof is finished.
where o(c) represents a function depending on R, ω, c, t 0 , t 1 that tends to zero uniformly for ω ∈ I as c → 0.
Proof. Let h(t, s) be the function defined in (4.5) on the the strip Ω = {(t, s) ∈ R 2 : 0 < t− s < σ}, with ε, σ and c given as in Proposition 4.3. Computations show that where R t := R(t) and R s := R(s), and we recall from (3.16) that Consider the function where we have used that As in Lemma 6.2, since t 1 (t, K, c)−t and t−t −1 (t, K, c) are uniformly bounded for c fixed, we can write Using these formulas in the expression of a c (t, K) for t =t, sinceṘt = 0 we get from standard computations where o(c) is as in Lemma 6.2.
The following result is crucial for the proof of Theorem 2.7.
Proposition 6.5. Suppose that R(t) ∈ R. Then, there exists c 0 such that for every c ∈ (0, c 0 ) and ω ∈ I ⊂ Ξ R the extended mapP does not admit any invariant curve with rotation number ω.
Proof. Suppose by contradiction that for c → 0, the mapP has an invariant curve Γ with rotation number ω ∈ I. By (B.2), for every orbit {(t n , K n )} on Γ we have that By the choice of β, for every n ∈ Z, that is (t n , t n+1 ) ∈ Ω β for every n ∈ Z. Hence the dynamics on the invariant curve are given by the billiard map P . By condition (iii) in Definition 2.2, there exists a point (t,K) ∈ Γ such thaṫ Open Access. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons. org/licenses/by/4.0/.
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Remark A.2. Notice that for ε = 1 the class R is empty. From condition (iii) and the definition of σ, that is impossible.
Finally it is immediate from the hypothesis that δ < 1. Let us start with the computation of σ. Using (A.1), we have that We note that d 2 using the fact that α ∈ (1, √ 2) one can show that This gives condition (i).

Since by (A.3)
M > 8π 2 δ(k 2 + 1) that gives the first inequality in (ii). To prove the second inequality, we note that the first inequality in (A.4) gives so that we are done if Looking at the asymptotic behaviour as M → +∞ of the left-and right-hand side of the previous inequality, we find that the condition is equivalent to Hence, the inequality is satisfied for M large enough if 1 2π(k 2 + 1) − (k + 1) < α 2 2π(k 2 + 1) + α 2 (k + 1) .

B. Some Results of Aubry-Mather Theory
In this section, we gather the results from Aubry-Mather theory that are used in the paper. For the proofs, we refer to [3,4,11,27,28].
In the following, we will tacitly consider the lift of S to the universal cover R 2 of A where x ∈ R,x(x + 1, y) =x(x, y) + 1 andȳ(x + 1, y) =ȳ(x, y). With some abuse, we will use the same notation for S and its lift, and the correct interpretation should be clear from the context.
We suppose that S is exact symplectic and twist. The exact symplectic condition requires the existence of a C 2 function V : Σ → R such that ydx − ydx = dV (x, y) in Σ, and the (positive) twist condition reads ∂x ∂y > 0 in Σ.
A negative twist condition would give analogous results. If Σ = A, we also suppose that S preserves the ends of the cylinder that is y → ±∞ as y → ±∞ uniformly in x, and twists each end infinitely that is x − x → ±∞ as y → ±∞ uniformly in x.
Note that the exact symplectic condition implies that S is orientation preserving and preserves the two-form dy ∧ dx.
For this class of maps, the following result is well known [4,28]. In the following, we denote the partial derivative of h with respect to the i-th variable by h i . Remark B.2. If Σ = A, the fact that S preserves and twists each end infinitely implies that Ω = R 2 . The condition h 12 (x,x) < 0 is related to the twist condition. Actually, the twist implies that we can write y = y(x,x) and one gets that The function h (or h ) is called generating function and gives an equivalent implicit definition of the diffeomorphism S. From this proposition, one has that a sequence (x n , y n ) n∈Z such that (x n , y n ) ∈ Σ for every n ∈ Z is an orbit of S if and only if for every n ∈ Z one has (x n , x n+1 ) ∈ Ω and h 2 (x n−1 , x n ) + h 1 (x n , x n+1 ) = 0, y n = −h 1 (x n , x n+1 ). (B.1) From now on, we consider the case Σ = A. Actually, the following extension result (see, for example, [22,28]) guarantees that we can always extend an exact symplectic diffeomorphism defined on a strip to one defined on the cylinder.

Lemma B.3.
Let h be a C 2 generating function defined on Ω = (x,x) ∈ R 2 : a ≤x − x ≤ b such that h 12 ≤ δ < 0 on Ω. Then there exists a generating functionh defined on R 2 such that h =h on Ω andh ≤ δ < 0 on R 2 . Moreover, It is well known that solutions of (B.1) (and hence orbits of S) are in 1-1 correspondence with stationary points of H k with respect to variations fixing the endpoints x , x k . In the following, we are interested in minimal orbits, i.e. orbits (x n , y n ) n∈Z of S such that for every pair of integers h < k and for every sequence of real numbers (x * n ) ≤n≤k such that x * = x h and x * k = x k , it holds H k (x , . . . , x k ) ≤ H k (x * , . . . , x * k ). Moreover, we recall that an orbit (x n , y n ) n∈Z of S has rotation number ω ∈ R if lim n→∞ x n n = ω.
It is well known that minimal orbits are monotone, that is only one of the following is satisfied: x n < x n+1 for every n ∈ Z, x n = x n+1 for every n ∈ Z, x n > x n+1 for every n ∈ Z.
Moreover, if it has rotation number ω, then it satisfies the following estimate for every n, m ∈ Z: Finally we recall that an invariant set of S is said to be minimal and with rotation number ω if it is made of minimal orbits with rotation number ω and that the term invariant curve of S refers to a curve Γ ⊂ Σ homotopic to {(x, y) ∈ A : y = k, for some k ∈ R} and such that S(Γ) = Γ. The following theorem gives the existence of minimal orbits with rotation number.
Theorem B.4 [ [4,27]]. Let h : R 2 → R be a C 2 generating function such that (i) h(x + 1,x + 1) = h(x,x) in R 2 , (ii) h 12 (x,x) ≤ δ < 0 in R 2 and let S be the corresponding diffeomorphism. For a fixed ω ∈ R • if ω = p/q ∈ Q, then there exists a minimal orbit (x n , y n ) n∈Z of S such that (x n+q , y n+q ) = (x n + p, y n ) • if ω ∈ R \ Q, then there exists a minimal invariant set M ω of rotation number ω such that M ω is the graph of a Lipschitz function u : π(M ω ) → R. Moreover, M ω is either an invariant curve or a Cantor set.
The following corollary gives an equivalent interpretation of the result and has been proven in [26] (see also [23]).
For irrational rotation numbers ω, Theorem B.4 leaves open the possibility for the minimal set M ω to be an invariant curve or not. To prove what is the case for a given ω is of fundamental importance to prove the existence of chaotic motion for the diffeomorphism S. We recall the following result by Forni. Let us fix ω ∈ R \ Q and denote by σ ω the unique S-invariant ergodic Borel probability measure supported on M ω . Theorem B.6 [ [11]]. Let S be a C 2 diffeomorphism of the cylinder A as in Theorem B.4. Suppose that S does not admit any invariant curve of rotation number ω. Then there exists an S-invariant ergodic Borel probability measure μ ω with positive metric entropy. Moreover, μ ω can be chosen arbitrarily close to σ ω in the sense of the weak topology on the space of compactly supported Borel probability measures on A.
Finally we recall a result to prove whether the set M ω is an invariant curve or not.
Proposition B.7 [ [25]]. Let S be a C 2 diffeomorphism of the cylinder A as in Theorem B.4, and let Γ be an invariant curve of S. Then (i) Γ is a minimal set and each orbit on Γ has the same rotation number; (ii) for any orbit (x n , y n ) on Γ, it holds a(x n−1 , x n , x n+1 ) := h 22 (x n−1 , x n ) + h 11 (x n , x n+1 ) > 0 , ∀ n ∈ Z .