Pinball billiards with dominated splitting

Abstract

We study the dynamics of a type of nonconservative billiards where the ball is "kicked" by the wall giving a new impulse in the direction of the normal. For different types of billiard tables we study the existence of attractors with dominated splitting.

Microorganism billiards in closed plane curves

“…We point out the ramifications of these orientation reversals and when the dynamics is completely or only partially encapsulated in the appropriate one-dimensional map F (q), where q is the arclength parameter for the table boundary. The derivative of this map is given [8,9] by…”

“…Current extensions to well-studied billiard problems include modifications of the table geometry, the shape of the inter-collision trajectories, and the rule for generating a new trajectory upon contact with the table boundary. In this direction, recent attention has been paid to aspecular reflection laws, especially in dissipative billiard systems commonly referred to as pinball billiards [8][9][10][11] and slap maps [12][13][14], as well as to aspecular reflection laws arising from other physical effects [15].…”

Dissipative outer billiards: a case study

“…By a recent result of Culter, every outer polygonal billiard has a periodic orbit [31]. A detailed knowledge of the periodic points of T λ for λ < 1 may help obtain information on periodic points when λ = 1 by studying the limit λ → 1; (2) the dissipative polygonal inner billiard exhibits chaotic attractors with SRB measures [1,8,9,22]. It would be instructive to compare these phenomena with the dynamics exhibited by dissipative polygonal outer billiards; (3) by a result of Bruin and Deane, almost every piecewise contraction is asymptotically periodic [4].…”

Limit sets of convex non-elastic billiards

“…In [19], non conservative billiards were introduced by a modification of the reflection rule and the existence of limit sets with a dominated splitting was demonstrated, for a wide class of dispersing and semidispersing billiards and of billiards with focusing components, but the dynamical type of the attractors was not investigated. In [2] models were studied numerically and different attractors, periodic and chaotic, are presented.…”

“…The maps we consider here are more general than the pinball billiards introduced in [19], as the perturbation of the angle is not necessarily biased to the normal direction.…”