Polygonal Billiards with Small Obstacles

The technique of unfolding a polygonal billiard table is used to answer certain questions concerning the illumination problem. The main problem addressed is how many point obstacles would suffice to block any billiard path between two points of the polygon. The answer can then be generalized from point obstacles to small -neighborhoods of points.     

Escape Rates and Physically Relevant Measures for Billiards with Small Holes

We study the billiard map corresponding to a periodic Lorentz gas in 2-dimensions in the presence of small holes in the table. We allow holes in the form of open sets away from the scatterers as well as segments on the boundaries of the scatterers. For a large class of smooth initial distributions, we establish the existence of a common escape rate and normalized limiting distribution. This limiting distribution is conditionally invariant and is the natural analogue of the SRB measure of a closed system. Finally, we prove that as the size of the hole tends to zero, the limiting distribution converges to the smooth invariant measure of the billiard map.     

Generalized Physical and SRB Measures for Hyperbolic Diffeomorphisms

In this paper we introduce the notion of generalized physical and SRB measures. These measures naturally generalize classical physical and SRB measures to measures which are supported on invariant sets that are not necessarily attractors. We then perform a detailed case study of these measures for hyperbolic Hènon maps. For this class of systems we are able to develop a complete theory about the existence, uniqueness, finiteness, and properties of these natural measures. Moreover, we derive a classification for the existence of a measure of full dimension. We also consider general hyperbolic surface diffeomorphisms and discuss possible extensions of, as well as the differences to, the results for Hènon maps. Finally, we study the regular dependence of the dimension of the generalized physical/SRB measure on the diffeomorphism. For the proofs we apply various techniques from smooth ergodic theory including the thermodynamic formalism. 2000 Mathematics Subject Classification. Primary: 37C45, 37D20, 37D35, Secondary: 37A35, 37E30     

Non-Stationary Non-Uniform Hyperbolicity: SRB Measures for Dissipative Maps

We prove the existence of SRB measures for diffeomorphisms where a positive volume set of initial conditions satisfy an “effective hyperbolicity” condition that guarantees certain recurrence conditions on the iterates of Lebesgue measure. We give examples of systems that do not admit a dominated splitting but can be shown to have SRB measures using our methods.     

Entropy Production for a Class of Inverse SRB Measures

We study the entropy production for inverse SRB measures for a class of hyperbolic folded repellers presenting both expanding and contracting directions. We prove that for most such maps we obtain strictly negative entropy production of the respective inverse SRB measures. Moreover we provide concrete examples of hyperbolic folded repellers where this happens.     

Limit Theorems for Dispersing Billiards with Cusps

Dispersing billiards with cusps are deterministic dynamical systems with a mild degree of chaos, exhibiting “intermittent” behavior that alternates between regular and chaotic patterns. Their statistical properties are therefore weak and delicate. They are characterized by a slow (power-law) decay of correlations, and as a result the classical central limit theorem fails. We prove that a non-classical central limit theorem holds, with a scaling factor of \({\sqrt{n\log n}}\) replacing the standard \({\sqrt{n}}\) . We also derive the respective Weak Invariance Principle, and we identify the class of observables for which the classical CLT still holds.     

Soft Billiards with Corners

We develop a framework for dealing with smooth approximations to billiards with corners in the two-dimensional setting. Let a polygonal trajectory in a billiard start and end up at the same billiard's corner point. We prove that smooth Hamiltonian flows which limit to this billiard have a nearby periodic orbit if and only if the polygon angles at the corner are acceptable. The criterion for a corner polygon to be acceptable depends on the smooth potential behavior at the corners, which is expressed in terms of a scattering function. We define such an asymptotic scattering function and prove the existence of it, explain how can it be calculated and predict some of its properties. In particular, we show that it is non-monotone for some potentials in some phase space regions. We prove that when the smooth system has a limiting periodic orbit it is hyperbolic provided the scattering function is not extremal there. We then prove that if the scattering function is extremal, the smooth system has elliptic periodic orbits limiting to the corner polygon, and, furthermore, that the return map near these periodic orbits is conjugate to a small perturbation of the Hénon map and therefore has elliptic islands. We find from the scaling that the island size is typically algebraic in the smoothing parameter and exponentially small in the number of reflections of the polygon orbit.     

Dispersing Billiards with Moving Scatterers

We propose a model of Sinai billiards with moving scatterers, in which the locations and shapes of the scatterers may change by small amounts between collisions. Our main result is the exponential loss of memory of initial data at uniform rates, and our proof consists of a coupling argument for non-stationary compositions of maps similar to classical billiard maps. This can be seen as a prototypical result on the statistical properties of time-dependent dynamical systems.     

Measures of Information and Error Laws

The logarithm of joint error densities for themost common means are shown to be proportional to thedifference of two weighted means which discriminatebetween a complete, nonuniform probability distribution and the uniform distribution. The difference inthe weighted means is related to a new Shannon-typeinequality for the discrimination between twoprobability distributions. Measures of the distancebetween the two distributions are determined, and a newstatistic, comparable to ², is derivedfrom a first-order approximatiom of the directeddivergence. Comparison is made between the error lawsand the method of maximum likelihood.     

Improved Estimates for Correlations in Billiards

We consider several classes of chaotic billiards with slow (polynomial) mixing rates, which include Bunimovich’s stadium and dispersing billiards with cusps. In recent papers by Markarian and the present authors, estimates on the decay of correlations were obtained that were sub-optimal (they contained a redundant logarithmic factor). We sharpen those estimates by removing that factor.     

Rotation Sets of Billiards with One Obstacle

We investigate the rotation sets of billiards on the m-dimensional torus with one small convex obstacle and in the square with one small convex obstacle. In the first case the displacement function, whose averages we consider, measures the change of the position of a point in the universal covering of the torus (that is, in the Euclidean space), in the second case it measures the rotation around the obstacle. A substantial part of the rotation set has usual strong properties of rotation sets.The first author was partially supported by NSF grant DMS 0456748.The second author was partially supported by NSF grant DMS 0456526.The third author was partially supported by NSF grant DMS 0457168.     

Semi-Dispersing Billiards with an Infinite Cusp I

 Let 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 conditions on f, we prove that the billiard flow in Q has a hyperbolic structure and, for some examples, that it is also ergodic. This is done using the cross section corresponding to collisions with the dispersing part of the boundary. The relevant invariant measure for this Poincaré section is infinite, whence the need to surpass the existing results, designed for finite-measure dynamical systems. Received: 1 May 2002 / Accepted: 13 May 2002 Published online: 22 August 2002     

Ergodic Directions for Billiards in a Strip with Periodically Located Obstacles

We study the size of the set of ergodic directions for the directional billiard flows on the infinite band ${\mathbb{R}\times [0,h]}$ with periodically placed linear barriers of length 0 < λ < h. We prove that the set of ergodic directions is always uncountable. Moreover, if λ/h ∈ (0, 1) is rational, the Hausdorff dimension of the set of ergodic directions is greater than 1/2. In both cases (rational and irrational) we construct explicitly some sets of ergodic directions.     

Generalizations of SRB Measures to Nonautonomous,Random, and Infinite Dimensional Systems

We review some developments that are direct outgrowths of, or closely related to, the idea of SRB measures as introduced by Sinai, Ruelle and Bowen in the 1970s. These new directions of research include the emergence of strange attractors in periodically forced dynamical systems, random attractors in systems defined by stochastic differential equations, SRB measures for infinite dimensional systems including those defined by large classes of dissipative PDEs, quasi-static distributions for slowly varying time-dependent systems, and surviving distributions in leaky dynamical systems.     

Dispersing Billiards with Cusps: Slow Decay of Correlations

Dispersing billiards introduced by Sinai are uniformly hyperbolic and have strong statistical properties (exponential decay of correlations and various limit theorems). However, if the billiard table has cusps (corner points with zero interior angles), then its hyperbolicity is nonuniform and statistical properties deteriorate. Until now only heuristic and experimental results existed predicting the decay of correlations as . We present a first rigorous analysis of correlations for dispersing billiards with cusps.