Gianluigi Del Magno

Chaos in the square billiard with a modified reflection law

The purpose of this paper is to study the dynamics of a square billiard with a non-standard reflection law such that the angle of reflection of the particle is a linear contraction of the angle of incidence. We present numerical and analytical arguments that the nonwandering set of this billiard decomposes into three invariant sets, a parabolic attractor, a chaotic attractor, and a set consisting of several horseshoes. This scenario implies the positivity of the topological entropy of the billiard, a property that is in sharp contrast with the integrability of the square billiard with the standard reflection law.

An infinite step billiard

A class of non-compact billiards is introduced, namely the infinite step billiards, i.e. systems of a point particle moving freely in the domain , with elastic reflections on the boundary; here , and . After describing some generic ergodic features of these dynamical systems, we turn to a more detailed study of the example . Playing an important role in this case are the so-called escape orbits, that is, orbits going to monotonically in the X-velocity. A fairly complete description of them is given. This enables us to prove some results concerning the topology of the dynamics on the billiard.

Dissipative outer billiards: a case study

We study dissipative polygonal outer billiards, i.e. outer billiards about convex polygons with a contractive reflection law. We prove that dissipative outer billiards about any triangle and the square are asymptotically periodic, i.e. they have finitely many global attracting periodic orbits. A complete description of the bifurcations of the periodic orbits as the contraction rates vary is given. For the square billiard, we also show that the asymptotic periodic behaviour is robust under small perturbations of the vertices and the contraction rates. Finally, we describe some numerical experiments suggesting that dissipative outer billiards about regular polygon are generically asymptotically periodic.

ERGODICITY OF POLYGONAL SLAP MAPS

Polygonal slap maps are piecewise affine expanding maps of the interval obtained by projecting the sides of a polygon along their normals onto the perimeter of the polygon. These maps arise in the study of polygonal billiards with non-specular reflection laws. We study the absolutely continuous invariant probabilities (acips) of the slap maps for several polygons, including regular polygons and triangles. We also present a general method for constructing polygons with slap maps with more than one ergodic acip.

Ergodicity of a class of truncated elliptical billiards

We consider a class of billiard tables obtained by intersecting elliptical domains x2/a2 + y2?1,a>1 with horizontal strips |y|?h<1. The boundary of these tables consists of two elliptical arcs connected by two parallel straight segments. We prove that the billiards in these tables have non-vanishing Lyapunov exponents for h<min (1/a,1/(2)1/2), and are ergodic for h<1/(1 + a2)1/2.

A local ergodic theorem for non-uniformly hyperbolic symplectic maps with singularities

In this paper, we prove a criterion for the local ergodicity of non-uniformly hyperbolic symplectic maps with singularities. Our result is an extension of a theorem of Liverani and Wojtkowski.

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 C2 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 C2 with finitely many intersection points.

Polygonal Billiards with Strongly Contractive Reflection Laws: A Review of Some Hyperbolic Properties

We provide an overview of recent results concerning the dynamics of polygonal billiards with strongly contractive reflection laws.