Markov Towers and Stochastic Properties of Billiards

Markov partitions work most efficiently for Anosov systems or for Axiom A systems. 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.