Roberto Markarian

Billiards with Pesin region of measure one

We give a description of a large class of plane billiards with Pesin region of measure one. Open conditions including properly those founded by Wojtkowski [W1] forC4 focusing boundaries are obtained. Lyapunovs forms, introduced by Lewowicz, are used.

Ergodic properties of Anosov maps with rectangular holes

We study Anosov diffeomorphisms on manifolds in which some ‘holes’ are cut. The points that are mapped into those holes disappear and never return. The holes studied here are rectangles of a Markov partition. Such maps generalize Smales horseshoes and certain open billiards. The set of nonwandering points of a map of this kind is a Cantor-like set calledrepeller. We construct invariant and conditionally invariant measures on the sets of nonwandering points. Then we establish ergodic, statistical, and fractal properties of those measures.

Conditionally invariant measures for Anosov maps with small holes

We study Anosov diffeomorphisms on surfaces in which some small ‘holes’ are cut. The points that are mapped into those holes disappear and never return. We assume that the holes are arbitrary open domains with piecewise smooth boundary, and their sizes are small enough. The set of points whose trajectories stay away from holes in the past is a Cantor-like union of unstable fibers. We establish the existence and uniqueness of a conditionally invariant measure on this set, whose conditional distributions on unstable fibers are smooth. This generalizes previous works by Pianigiani, Yorke, and others. AMS classification numbers: 58F12, 58F15, 58F11

Invariant measures for Anosov maps with small holes

We study Anosov diffeomorphisms on surfaces with small holes. The points that are mapped into the holes disappear and never return. In our previous paper [6] we proved the existence of a conditionally invariant measure μ+. Here we show that the iterations of any initially smooth measure, after renormalization, converge to μ+. We construct the related invariant measure on the repeller and prove that it is ergodic and K-mixing. We prove the escape rate formula, relating the escape rate to the positive Lyapunov exponent and the entropy. AMS classification numbers: 58F12, 58F15, 58F11

Open billiards: invariant and conditionally invariant probabilities on Cantor sets

Billiards are the simplest models for understanding the statistical theory of the dynamics of a gas in a closed compartment. We analyze the dynamics of a class of billiards (the open billiard on the plane) in terms of invariant and conditionally invariant probabilities. The dynamical system has a horseshoe structure. The stable and unstable manifolds are analytically described. The natural probability

Chaotic Properties of the Elliptical Stadium

\mu

New ergodic billiards: exact results

is invariant and has support in a Cantor set. This probability is the conditional limit of a conditional probability

STATIC AND TIME-DEPENDENT PERTURBATIONS OF THE CLASSICAL ELLIPTICAL BILLIARD

\mu _F

Anosov maps with rectangular holes. Nonergodic cases

that has a density with respect to the Lebesgue measure. A formula relating entropy, Lyapunov exponent, and Hausdorff dimension of a natural probability

Pinball billiards with dominated splitting

\mu