Related papers: Proving The Ergodic Hypothesis for Billiards With …

The connected configuration space of a so called cylindric billiard system is a flat torus minus finitely many spherical cylinders. The dynamical system describes the uniform motion of a point particle in this configuration space with…

We consider the billiard dynamics in a non-compact set of R^d that is constructed as a bi-infinite chain of translated copies of the same d-dimensional polytope. A random configuration of semi-dispersing scatterers is placed in each copy.…

The Local Ergodic Theorem (also known as the `Fundamental Theorem') gives sufficient conditions under which a phase point has an open neighborhood that belongs (mod 0) to one ergodic component. This theorem is a key ingredient of many…

We consider the billiard flow of elastically colliding hard balls on the flat $\nu$-torus ($\nu\ge 2$), and prove that no singularity manifold can even locally coincide with a manifold describing future non-hyperbolicity of the…

Dynamical billiards, or the behavior of a particle traveling in a planar region $D$ undergoing elastic collisions with the boundary, has been extensively studied and is used to model many physical phenomena such as a Boltzmann gas. Of…

We consider classical billiards in plane, connected, but not necessarily bounded domains. The charged billiard ball is immersed in a homogeneous, stationary magnetic field perpendicular to the plane. The part of dynamics which is not…

Let $f: [0, +\infty) \to (0, +\infty)$ be a sufficiently smooth convex function, vanishing at infinity. Consider the planar domain $Q$ delimited by the positive $x$-semiaxis, the positive $y$-semiaxis, and the graph of $f$. Under certain…

We consider the outer billiards map with contraction outside polygons. We construct a 1-parameter family of systems such that each system has an open set in which the dynamics is reduced to that of a piecewise contraction on the interval.…

We consider the system of $N$ ($\ge2$) hard balls with masses $m_1,...,m_N$ and radius $r$ in the flat torus $\Bbb T_L^\nu=\Bbb R^\nu/L\cdot\Bbb Z^\nu$ of size $L$, $\nu\ge3$. We prove the ergodicity (actually, the Bernoulli mixing…

We demonstrate that the free motion of any two-dimensional rigid body colliding elastically with two parallel, flat walls is equivalent to a billiard system. Using this equivalence, we analyze the integrable and chaotic properties of this…

In the ergodic theory of semi-dispersing billiards the Local Ergodic Theorem, proved by Chernov and Sinai in 1987, plays a central role. So far, all existing proofs of this theorem had to use an annoying global hypothesis, namely the almost…

We prove that flow of a generic geodesic on a flat surface with finite holonomy group is ergodic. We use this result to prove that flows of generic billiards on certain flat surfaces with boundary are also ergodic.

We prove polynomial upper bounds for the deviation of ergodic averages for the straight line flow on every translation surface in almost every direction, in particular for those surfaces arising from rational polygonal billiards.

In this paper, we show that billiard orbits in rational polygons and geodesics on translation surfaces exhibit super-fast spreading, an optimal time-quantitative majority property about the corresponding linear flow that implies uniformity…

We show ergodicity of (asymmetric) lemon billiards, billiard tables that are the intersection of two circles of which one contains the centers of both. These do not satisfy the Wojtkowski criteria for hyperbolicity, but we establish…

In a previous paper we defined a class of non-compact polygonal billiards, the infinite step billiards: to a given decreasing sequence of non-negative numbers $\{p_{n}$, there corresponds a table $\Bi := \bigcup_{n\in\N} [n,n+1] \times…

Any smooth surface in R^3 may be flattened along the z-axis, and the flattened surface becomes close to a billiard table in R^2 . We show that, under some hypotheses, the geodesic flow of this surface converges locally uniformly to the…

We study, by numerical simulations and semi-rigorous arguments, a two-parameter family of convex, two-dimensional billiard tables, generalizing the one-parameter class of oval billiards of Benettin--Strelcyn [Phys. Rev. A 17, 773 (1978)].…

Motivated by the study of billiards in polyhedra, we study linear flows in a family of singular flat $3$-manifolds which we call translation prisms. Using ideas of Furstenberg and Veech, we connect results about weak mixing properties of…

The present work consists of a numerical study of the dynamics of irrational polygonal billiards. Our contribution reinforces the hypothesis that these systems could be Strongly Mixing, although never demonstrably chaotic, and discuss the…