Markov Approximations and Statistical Properties of Billiards

Abstract

However, for hyperbolic dynamical systems which are either singular or whose hyperbolicity is nonuniform, the construction of a Markov partition, which in these cases is necessarily countable, is a rather delicate issue even when such a construction exists. An additional problem is the use of a countable Markov partition for proving probabilistic statements. For a wide class of hyperbolic systems, L. S. Young, in 1998, constructed so called Markov towers, which she could apply successfully to establish nice, for instance, exponential correlation decay, and, moreover, as a consequence, a central limit theorem. The aim of this survey is twofold. First we show how the Markov tower construction is applicable for obtaining finer stochastic properties, like a local limit theorem of probability theory. Here the fundamental method is the study of the spectrum of the Fourier transform of the Perron-Frobenius operator. These ideas and results are applicable to all systems Young has been considering. Second, we survey the problem of recurrence of the planar Lorentz process. As an application of the results from the first part, we obtain a dynamical proof of recurrence for the finite horizon case. Here basically different proofs were given by K. Schmidt, in 1998, and J.-P. Conze, in 1999. As another application we can also treat the infinite horizon case, where already the global limit theorem is absolutely novel. It is not a central one, the scaling is √ n log n in contrast to the classical √ n one. Beyond thus giving a rigorous proof for earlier heuristic ideas of P. Bleher, which used three delicate and hard hypotheses, we can also a) verify the local version of this limit theorem for the free flight function and b) prove the recurrence of the planar Lorentz process in the infinite horizon case.

Spherical billiards with almost complete escape

“…2 Mathematical billiards are of interest for a diverse collection of examples of dynamical systems (depending on the cavity shape). [3][4][5][6] Examples of dynamics include chaotic (e.g., Brownian motion, Lorentz flows, and the Sinai billiard 7 ), intermittent, e.g., the drive-belt stadium billiard, 8 and regular. 2,[9][10][11] We will now consider open dynamical systems, formed by the introduction of a small hole in the boundary or in the interior allowing us to probe their internal dynamical nature.…”

Diffusion Rate of Windtree Models and Lyapunov Exponents

“…In parallel a similar model with smooth convex obstacles has been studied by a large amount of mathematicians throughout the twentieth century (see e.g. [BS81] or [SV04]). In this case, the billiards satisfy some hyperbolicity property and the behaviour of its flow is closely related to a Brownian motion.…”

Two-Sided Shift Spaces Over Infinite Alphabets

“…Thermodynamical formalism for such shifts has also been well developed; see, for example, [5,15,16,20,23,25]. Seminal work of Young [30] defined Markov towers and used them to study hyperbolic systems; see also [31] and see [27] for a survey.…”