Billiard Map Research Articles

Bialy-Mironov type rigidity for centrally symmetric symplectic billiards

Abstract The aim of the present paper is to establish a Bialy–Mironov type rigidity for centrally symmetric symplectic billiards. For a centrally symmetric C 2 strongly-convex domain D with boundary ∂ D , assume that the symplectic billiard map has a (simple) continuous invariant curve δ ⊂ P of rotation number 1 / 4 (winding once around ∂ D ) and consisting only of 4-periodic orbits. If one of the parts between δ and each boundary of the phase-space is entirely foliated by continuous invariant closed (not null-homotopic) curves, then ∂ D is an ellipse. The differences with Birkhoff billiards are essentially two: it is possible to assume the existence of the foliation in one of the parts of the phase-space detected by the curve δ, and the result is obtained by tracing back the problem directly to the totally integrable case.

On the role of the surface geometry in convex billiards

Abstract This work presents a framework for billiards in convex domains on two dimensional Riemannian manifolds. These domains are contained in connected, simply connected open subsets which are totally normal. In this context, some properties that have long been known for billiards on the plane are established. We prove the twist property of the billiard maps and establish some conditions for the existence and non-existence of rotational invariant curves. Although we prove that Lazutkin’s and Hubacher’s theorems are valid for general surfaces, we also find that Mather’s theorem does not apply to surfaces of non-negative curvature.

Birkhoff attractors of dissipative billiards

Abstract We study the dynamics of dissipative billiard maps within planar convex domains. Such maps have a global attractor. We are interested in the topological and dynamical complexity of the attractor, in terms both of the geometry of the billiard table and of the strength of the dissipation. We focus on the study of an invariant subset of the attractor, the so-called Birkhoff attractor. On the one hand, we show that for a generic convex table with ‘pinched’ curvature, the Birkhoff attractor is a normally contracted manifold when the dissipation is strong. On the other hand, for a mild dissipation, we prove that, generically, the Birkhoff attractor is complicated, both from the topological and the dynamical points of view.

Symbolic Dynamics for Nonuniformly Hyperbolic Maps with Singularities in High Dimension

We construct Markov partitions for non-invertible and/or singular nonuniformly hyperbolic systems defined on higher dimensional Riemannian manifolds. The generality of the setup covers classical examples not treated so far, such as geodesic flows in closed manifolds, multidimensional billiard maps, and Viana maps, and includes all the recent results of the literature. We also provide a wealth of applications.

Random circular billiards on surfaces of constant curvature: Pseudo integrability and mixing

Given a random map [Formula: see text], we define a random billiard map on a surface of constant curvature (Euclidean plane, hyperbolic plane, or the sphere). The Liouville measure is invariant for this billiard map. Finally, we show some dynamical properties such as ergodicity in the case of random circular billiards.

Outer billiards in the spaces of oriented geodesics of the three‐dimensional space forms

AbstractLet be the three‐dimensional space form of constant curvature , that is, Euclidean space , the sphere , or hyperbolic space . Let be a smooth, closed, strictly convex surface in . We define an outer billiard map on the four‐dimensional space of oriented complete geodesics of , for which the billiard table is the subset of consisting of all oriented geodesics not intersecting . We show that is a diffeomorphism when is quadratically convex. For , has a Kähler structure associated with the Killing form of . We prove that is a symplectomorphism with respect to its fundamental form and that can be obtained as an analogue to the construction of Tabachnikov of the outer billiard in defined in terms of the standard symplectic structure. We show that does not preserve the fundamental symplectic form on associated with the cross product on , for . We initiate the dynamical study of this outer billiard in the hyperbolic case by introducing and discussing a notion of holonomy for periodic points.

Rates of mixing for the measure of maximal entropy of dispersing billiard maps

AbstractIn a recent work, Baladi and Demers constructed a measure of maximal entropy for finite horizon dispersing billiard maps and proved that it is unique, mixing and moreover Bernoulli. We show that this measure enjoys natural probabilistic properties for Hölder continuous observables, such as at least polynomial decay of correlations and the Central Limit Theorem. The results of Baladi and Demers are subject to a condition of sparse recurrence to singularities. We use a similar and slightly stronger condition, and it has a direct effect on our rate of decay of correlations. For billiard tables with bounded complexity (a property conjectured to be generic), we show that the sparse recurrence condition is always satisfied and the correlations decay at a super‐polynomial rate.

Effective rigidity away from the boundary for centrally symmetric billiards

Abstract In this paper, we study centrally symmetric Birkhoff billiard tables. We introduce a closed invariant set $\mathcal {M}_{\mathcal {B}}$ consisting of locally maximizing orbits of the billiard map lying inside the region $\mathcal {B}$ bounded by two invariant curves of $4$ -periodic orbits. We give an effective bound from above on the measure of this invariant set in terms of the isoperimetric defect of the curve. The equality case occurs if and only if the curve is a circle.

An Upper Bound on Topological Entropy of the Bunimovich Stadium Billiard Map

We show that the topological entropy of the billiard map in a Bunimovich stadium is at most log (3.49066).

On infinitely many foliations by caustics in strictly convex open billiards

AbstractReflection in a strictly convex bounded planar billiard acts on the space of oriented lines and preserves a standard area form. A caustic is a curve C whose tangent lines are reflected by the billiard to lines tangent to C. The famous Birkhoff conjecture states that the only strictly convex billiards with a foliation by closed caustics near the boundary are ellipses. By Lazutkin’s theorem, there always exists a Cantor family of closed caustics approaching the boundary. In the present paper, we deal with an open billiard, whose boundary is a strictly convex embedded (non-closed) curve $\gamma $ . We prove that there exists a domain U adjacent to $\gamma $ from the convex side and a $C^\infty $ -smooth foliation of $U\cup \gamma $ whose leaves are $\gamma $ and (non-closed) caustics of the billiard. This generalizes a previous result by Melrose on the existence of a germ of foliation as above. We show that there exists a continuum of above foliations by caustics whose germs at each point in $\gamma $ are pairwise different. We prove a more general version of this statement for $\gamma $ being an (immersed) arc. It also applies to a billiard bounded by a closed strictly convex curve $\gamma $ and yields infinitely many ‘immersed’ foliations by immersed caustics. For the proof of the above results, we state and prove their analogue for a special class of area-preserving maps generalizing billiard reflections: the so-called $C^{\infty }$ -lifted strongly billiard-like maps. We also prove a series of results on conjugacy of billiard maps near the boundary for open curves of the above type.

Topologically weakly mixing polygonal billiards

We show that in a typical polygon the billiard map as well as its associated subshift obtained by coding orbits by the sequence of sides they visit are topologically weakly mixing.

The wave trace and Birkhoff billiards

The purpose of this article is to develop a Hadamard–Riesz-type parametrix for the wave propagator in bounded planar domains with smooth, strictly convex boundary. This parametrix then allows us to rederive an oscillatory integral representation for the wave trace appearing in the work by Marvizi and Melrose (1982) and compute its principal symbol explicitly in terms of geometric data associated to the billiard map. This results in new formulas for the wave invariants. The order of the principal symbol, which appears to be inconsistent in the works by Marvizi and Melrose (1982) and Popov (1994), is also corrected. In those papers, the principal symbol was never actually computed and to our knowledge, this paper contains the first explicit formulas for the principal symbol of the wave trace. The wave trace formulas we provide are localized near both simple lengths corresponding to nondegenerate periodic orbits and degenerate lengths associated to one parameter families of periodic orbits tangent to a single rational caustic. Existence of a Hadamard–Riesz-type parametrix with explicit symbol and phase calculations in the interior appears to be new in the literature, with the exception of the author’s previous work [J. Geom. Anal. 31 (2021), 2238–2295] in the special case of elliptical domains. This allows us to circumvent the symbol calculus used by Duistermaat and Guillemin (1975) and Hezari and Zelditch (2012) when computing trace formulas, which are instead derived from integrating our explicit parametrix over the diagonal.

Locally maximising orbits for the non-standard generating function of convex billiards and applications

Given an exact symplectic map T of a cylinder with a generating function H satisfying the so-called negative twist condition, , we study the locally maximising orbits of T, that is, configurations which are local maxima of the action functional . We provide a necessary and sufficient condition for a configuration to be locally maximising. Using it, we consider a situation where T has two generating functions with respect to two different sets of symplectic coordinates. We suggest a simple geometric condition which guarantees that the set of locally maximising orbits with respect to both of these generating functions coincide. As the main application we show that the two generating functions for planar Birkhoff billiards satisfy this geometric condition. We apply it to get the following result: consider a centrally symmetric curve γ, for which the Birkhoff billiard map has a rotational invariant curve α of four-periodic orbits. We prove that a certain L 2-distance between γ and its ‘best approximating’ ellipse can be bounded from above in terms of the measure of the complement of the set filled by locally maximising orbits lying between α and the boundary of the phase cylinder. Moreover, this estimate is sharp, giving an effective version of a recent result on Birkhoff conjecture for centrally symmetric curves (Bialy and Mironov 2022 Ann. Math. 196 389–413). We also get a similar bound for arbitrary curves γ, that relates the measure of the complement of the set of locally maximising orbits with the L 2-distance between γ and its ‘best approximating’ circle.

Holography of geodesic flows, harmonizing metrics, and billiards' dynamics

For a traversing vector field $v$ on a compact $(n+1)$-manifold $X$ with boundary, we use closed $v$-invariant differential $n$-forms $\Theta$ to define measures $\mu_\Theta$ on the boundary $\partial X$, such that the $v$-flow generated causality map $C_v: \partial^+X(v) \to \partial^-X(v)$ preserves $\mu_\Theta$. In combination with a $\mu_\Theta$-preserving involution $\tau: \partial X \to \partial X$, which maps $\partial^-X(v)$ to $\partial^+X(v)$, the proto-billiard map $B_{v, \tau} := \tau \circ C_v$ is a measure-preserving transformation of $\partial^+X(v)$. We study the dynamical system $(\partial^+X(v), B_{v, \tau})$. Let $(M, g)$ be a Riemannian $n$-manifold with boundary, where $g$ is a non-trapping metric. Let $SM$ be the space of the spherical tangent to $M$ bundle, and $v^g$ the geodesic vector field on $SM$. We apply our treatment of $C_v$ and $(\partial^+X(v), B_{v, \tau})$ to the scattering maps $C_{v^g}$ and billiard maps $B_{v^g, \tau}$, where $\tau$ is generated by the reflection of unit tangent vectors with respect to $\partial M$. As a result, we getting a variety of holography theorems that tackle the inverse scattering problems for $C_{v^g}$ and theorems that describe the dynamics of $B_{v^g, \tau}$. Our main tool is the Lyapunov function $F: SM \to \mathbf R$ for $v^g$ and the metrics $g^\bullet$ on $SM$ that make $dF$ harmonic. For such metrics, we get isoperimetric inequalities $vol_{g^\bullet}(SM) \leq vol_{g^\bullet |}(\partial(SM))$. Assuming ergodicity of $B_{v^g, \tau}$ and non-trapping of $g$, we also get formulas for the average length of free geodesic segments in $M$.

On Some Invariants of Birkhoff Billiards Under Conjugacy

In the class of strictly convex smooth boundaries each of which has no strip around its boundary foliated by invariant curves, we prove that the Taylor coefficients of the “normalized” Mather’s \(\beta\)-function are invariant under \(C^{\infty}\)-conjugacies. In contrast, we prove that any two elliptic billiard maps are \(C^{0}\)-conjugate near their respective boundaries, and \(C^{\infty}\)-conjugate, near the boundary and away from a line passing through the center of the underlying ellipse. We also prove that, if the billiard maps corresponding to two ellipses are topologically conjugate, then the two ellipses are similar.

Eigenfunction asymptotics and spectral rigidity of the ellipse

Microlocal defect measures for Cauchy data of Dirichlet, resp. Neumann, eigenfunctions of an ellipse E are determined. We prove that, for any invariant curve for the billiard map on the boundary phase space B^* E of an ellipse, there exists a sequence of eigenfunctions whose Cauchy data concentrates on the invariant curve. We use this result to give a new proof that ellipses are infinitesimally spectrally rigid among C^\infty domains with the symmetries of the ellipse.

A random billiard map in the circle

We consider a random billiard map, the one in which the standard specular reflection rule is replaced by a random reflection given by a Markov operator. We exhibit an invariant measure for random billiards on general tables. In the special case of a circular table we show that almost every (random) orbit is dense in the boundary as well as in the circular ring formed between the circle boundary and the random caustic. We additionally prove Strong Knudsen's Law for a particular case of families of absolutely continuous measures with respect to Liouville. We also calculate the Lyapunov exponent for the random billiard map in the circle.

Billiard Tables with Rotational Symmetry

Abstract We generalize the following simple geometric fact: the only centrally symmetric convex curve of constant width is a circle. Billiard interpretation of the condition of constant width reads: a planar curve has constant width, if and only if, the Birkhoff billiard map inside the planar curve has a rotational invariant curve of $2$-periodic orbits. We generalize this statement to curves that are invariant under a rotation by angle $\frac {2\pi }{k}$, for which the billiard map has a rotational invariant curve of $k$-periodic orbits. Similar result holds true also for outer billiards and symplectic billiards. Finally, we consider Minkowski billiards inside a unit disc of Minkowski (not necessarily symmetric) norm that is invariant under a linear map of order $k\ge 3$. We find a criterion for the existence of an invariant curve of $k$-periodic orbits. As an application, we get rigidity results for all those billiards.

Expansiveness and Hyperbolicity in Convex Billiards

We say that a convex planar billiard table \(B\) is \(C^{2}\)-stably expansive on a fixed open subset \(U\) of the phase space if its billiard map \(f_{B}\) is expansive on the maximal invariant set \(\Lambda_{B,U}=\bigcap_{n\in\mathbb{Z}}f^{n}_{B}(U)\), and this property holds under \(C^{2}\)-perturbations of the billiard table. In this note we prove for such billiards that the closure of the set of periodic points of \(f_{B}\) in \(\Lambda_{B,U}\) is uniformly hyperbolic. In addition, we show that this property also holds for a generic choice among billiards which are expansive.

Almost sure invariance principle for random dynamical systems via Gouëzel's approach

We extend the spectral approach of S Gouëzel for the vector-valued almost sure invariance principle (ASIP) to certain classes of non-stationary sequences with a weaker control over the behaviour of the covariance matrices, assuming only linear growth. Then we apply this extension to obtain a quenched vector-valued ASIP for random perturbations of a fixed Anosov diffeomorphism as well as random perturbations of a billiard map associated to the periodic Lorentz gas. We also consider certain classes of random piecewise expanding maps.