Billiards in rational polygons: Periodic trajectories, symmetries, and d-stability

Periodic trajectories of billiards in rational polygons satisfying the Veech alternative, in particular, in right triangles with an acute angle of the form π/n with integern are considered. The properties under investigation include: symmetry of periodic trajectories, asymptotics of the number of trajectories whose length does not exceed a certain value, stability of periodic billiard trajectories under small deformations of the polygon. Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 66–75, July, 1997. Translated by V. N. Dubrovsky     

Degenerate billiards

In an ordinary billiard system, trajectories of a Hamiltonian system are elastically reflected after a collision with a hypersurface (scatterer). If the scatterer is a submanifold of codimension more than 1, we say that the billiard is degenerate. We study those trajectories of degenerate billiards that have an infinite number of collisions with the scatterer. Degenerate billiards appear as limits of systems with elastic reflections or as small-mass limits of systems with singularities in celestial mechanics. We prove the existence of trajectories of such systems that shadow the trajectories of the corresponding degenerate billiards. The proofs are based on a version of the method of an anti-integrable limit.     

Degenerate billiards in celestial mechanics

In an ordinary billiard trajectories of a Hamiltonian system are elastically reflected after a collision with a hypersurface (scatterer). If the scatterer is a submanifold of codimension more than one, we say that the billiard is degenerate. Degenerate billiards appear as limits of systems with singularities in celestial mechanics. We prove the existence of trajectories of such systems shadowing trajectories of the corresponding degenerate billiards. This research is motivated by the problem of second species solutions of Poincaré.     

Billiards and Nonholonomic Distributions

In this note, we consider billiards with full families of periodic orbits. It is shown that the construction of a convex billiard with a “rational” caustic (i.e., carrying only periodic orbits) can be reformulated as a problem of finding a closed curve tangent to an (N - 1)-dimensional distribution on a (2N - 1)-dimensional manifold. We describe the properties of this distribution, as well as some important consequences for billiards with rational caustics. A very particular application of our construction states that an ellipse can be infinitesimally perturbed so that any chosen rational elliptic caustic will persist. Bibliography: 13 titles.__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 300, 2003, pp. 56–64.     

Exponential Decay of Correlations in Multi-Dimensional Dispersing Billiards

We prove exponential decay of correlations for a “reasonable” class of multi-dimensional dispersing billiards. The scatterers are required to be C ³ smooth, the horizon is finite, there are no corner points. In addition, we assume subexponential complexity of the singularity set. Submitted: November 10, 2007. Accepted: May 23, 2008.     

W-flows on Weyl manifolds and Gaussian thermostats

We introduce W-flows, by modifying the geodesic flow on a Weyl manifold, and show that they coincide with the isokinetic dynamics. We establish some connections between negative curvature of the Weyl structure and the hyperbolicity of W-flows, generalizing in dimension 2 the classical result of Anosov on Riemannian geodesic flows. In higher dimensions we establish only weaker hyperbolic properties. We extend the theory to billiard W-flows and introduce the Weyl counterparts of Sinai billiards. We obtain that the isokinetic Lorentz gas with the constant external field E and scatterers of radius r, studied by Chernov, Eyink, Lebowitz and Sinai, is uniformly hyperbolic, if only r|E|<1, and this condition is sharp.     

Creating Transverse Homoclinic Connections in Planar Billiards

Given a planar billiard system containing stable and unstable manifolds that intersect nontransversely, we show how to make a local perturbation to the boundary that causes the intersection to become transverse. We apply these ideas to billiards inside an ellipse. Bibliography: 19 titles.__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 300, 2003, pp. 122–134.his revised version was published in June 2005 with corrections in the author affiliation.     

On 4-Reflective Complex Analytic Planar Billiards

The famous conjecture of Ivrii (Funct Anal Appl 14(2):98–106, 1980) says that in every billiard with infinitely-smooth boundary in a Euclidean space the set of periodic orbits has measure zero. In the present paper we study its complex analytic version for quadrilateral orbits in two dimensions, with reflections from holomorphic curves. We present the complete classification of 4-reflective complex analytic counterexamples: billiards formed by four holomorphic curves in the projective plane that have open set of quadrilateral orbits. This extends the author’s previous result Glutsyuk (Moscow Math J 14(2):239–289, 2014) classifying 4-reflective complex planar algebraic counterexamples. We provide applications to real planar billiards: classification of 4-reflective germs of real planar \(C^4\)-smooth pseudo-billiards; solutions of Tabachnikov’s Commuting Billiard Conjecture and the 4-reflective case of Plakhov’s Invisibility Conjecture (both in two dimensions; the boundary is required to be piecewise \(C^4\)-smooth). We provide a survey and a small technical result concerning higher number of complex reflections.     

A quantified Tauberian theorem for the Laplace-Stieltjes transform

We study the general rotation sets of open billiards in \({\mathbb {R}}^2\) for the observable given by the starting point of a given billiard trajectory. We prove that, for a class of open billiards, the general rotation set is equal to the polygon formed by the midpoints of the segments connecting the centers of the obstacles. Moreover, we provide an example to show that such a result may not hold in general.     

A billiards-like dynamical system for attacking chess pieces

We apply a one-dimensional discrete dynamical system originally considered by Arnol’d reminiscent of mathematical billiards to the study of two-move riders, a type of fairy chess piece. In this model, particles travel through a bounded convex region along line segments of one of two fixed slopes.We apply this dynamical system to characterize the vertices of the inside-out polytope arising from counting placements of nonattacking chess pieces and also to give a bound for the period of the counting quasipolynomial. The analysis focuses on points of the region that are on trajectories that contain a corner or on cycles of full rank, or are crossing points thereof.As a consequence, we give a simple proof that the period of the bishops’ counting quasipolynomial is 2, and provide formulas bounding periods of counting quasipolynomials for many two-move riders including all partial nightriders. We draw parallels to the theory of mathematical billiards and pose many new open questions.     

Regularity of Bunimovich’s stadia

Stadia are popular models of chaotic billiards introduced by Bunimovich in 1974. They are analogous to dispersing billiards due to Sinai, but their fundamental technical characteristics are quite different. Recently many new results were obtained for various chaotic billiards, including sharp bounds on correlations and probabilistic limit theorems, and these results require new, more powerful technical apparatus. We present that apparatus here, in the context of stadia, and prove “regularity” properties.      

The Distribution of Gaps for Saddle Connection Directions

Motivated by the study of billiards in polygons, we prove fine results for the distribution of gaps of directions of saddle connections on translation surfaces. As an application we prove that for almost every holomorphic differential ?? on a Riemann surface of genus g ?? 2 the smallest gap between saddle connection directions of length at most a fixed length decays faster than quadratically in the length. We also characterize the exceptional set: the decay rate is not faster than quadratic if and only if ?? is a lattice surface.     

Singular sets of planar hyperbolic billiards are regular

Many planar hyperbolic billiards are conjectured to be ergodic. This paper represents a first step towards the proof of this conjecture. The Hopf argument is a standard technique for proving the ergodicity of a smooth hyperbolic system. Under additional hypotheses, this technique also applies to certain hyperbolic systems with singularities, including hyperbolic billiards. The supplementary hypotheses concern the subset of the phase space where the system fails to be C ² differentiable. In this work, we give a detailed proof of one of these hypotheses for a large collection of planar hyperbolic billiards. Namely, we prove that the singular set and each of its iterations consist of a finite number of compact curves of class C ² with finitely many intersection points.     

Convex billiards

This article deals with (partly rather peculiar) properties of typical convex billiards. In particular, convex caustics, trajectories which terminate on the boundary and dense trajectories are investigated.Dedicated to Professor Curt Christian on the occasion of his 70th birthday     

Center manifolds for nonuniformly partially hyperbolic diffeomorphisms

We establish the existence of smooth invariant center manifolds for the nonuniformly partially hyperbolic trajectories of a diffeomorphism in a Banach space. This means that the differentials of the diffeomorphism along the trajectory admit a nonuniform exponential trichotomy. We also consider the more general case of sequences of diffeomorphisms, which corresponds to a nonautonomous dynamics with discrete time. In addition, we obtain an optimal regularity for the center manifolds: if the diffeomorphisms are of class C^k then the manifolds are also of class C^k. As a byproduct of our approach we obtain an exponential control not only for the trajectories on the center manifolds, but also for their derivatives up to order k.     

Connected components of the moduli spaces of Abelian differentials with prescribed singularities

Consider the moduli space of pairs (C,) where C is a smooth compact complex curve of a given genus and is a holomorphic 1-form on C with a given list of multiplicities of zeroes. We describe connected components of this space. This classification is important in the study of dynamics of interval exchange transformations and billiards in rational polygons, and in the study of geometry of translation surfaces.     

Periodic Billiard Trajectories in Smooth Convex Bodies

We consider billiard trajectories in a smooth convex body in \mathbbR^d{\mathbb{R}^{d}} and estimate the number of distinct periodic trajectories that make exactly p reflections per period at the boundary of the body. In the case of prime p we obtain the lower bound (d – 2)(p – 1) + 2, which is much better than the previous estimates.     

Topological Obstacles to the Realizability of Integrable Hamiltonian Systems by Billiards

Doklady Mathematics - We introduce the following classes of integrable billiards: elementary billiards, topological billiards, billiard books, billiards with a potential, with a magnetic field, and...     

Periodic reflecting paths in right triangles

A reflecting path in a Jordan curve has the ready physical analogies of a billiards ball bouncing off the cushions of a custom billiards table and of a laser beam reflecting off the mirrored boundary of a thin cavity. What does the curve tell us about the possible reflecting paths? Two types of reflecting paths are of perennial interest — periodic paths and dense paths. In 1927, G. D. Birkhoff applied a theorem of Poincaré to the billiards ball analogy to show that for anyk>1 and anyw<k/2, with (w, k)=1, a smooth convex curve admits at least two periodic reflecting paths ofk reflections per cycle and winding numberw. Unfortunately Birkhoff's proof does not extent to polygons, nor has any other method been found to yield such sweeping results. Various techniques have produced results for polygons-with-angles-commensurable-with-π and for acute triangles. It was not clear whether an arbitrary right triangle admitted periodic paths. Two simple constructions demonstrate the existence of periodic reflecting paths in right triangles. This paper explores these two constructions, focusing on the simplest result: Let β represent the smallest interior angle of some right triangle, and letN=[π/2β]?1. Then each point interior to the shorter leg of the right triangle lies on a unique reflecting path of (4k+2) reflections per cycle for allk=1,2,...,N?1; and each point interior to some subsegment of the shorter leg lies on a unique reflecting path of 4N+2 reflections per cycle.