The return time statistics for non-markovian maps

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Abstract proposal DMS-0602202The proposed work is on the connection between mixing properties, the decay of correlations and statistical properties of dynamical systems. There are three main areas:(i) We want to get a better understanding what happens to the return times for mixing maps near periodic orbits. Currently it is known that generic points have in the limit Poisson distributed return times. However near periodic orbits such regular behavior cannot be assumed and we intend to show that the limiting distribution is a compounded Poisson distribution where the compounded part is Bernoulli and determined by an `escape rate' near the periodic orbit.(ii) The second area of research is to determine the limiting distribution for return times for non-uniformly hyperbolic systems. This would allow us then to classify the long term statistical behavior for a considerably larger class of dynamical systems including some parabolic systems, `billiard type' systems and also higher dimensional maps similar to the Henon map which has been studied extensively and is technically very difficult to approach.(iii) The third major area in which we propose to do research is related to a famous theorem by Shannon, McMillan and Breiman that uses the decay rate of dynamical neighbourhoods to describe the entropy. We want to prove a Central Limit Theorem for a $\alpha$-mixing systems. This will considerably extend previous results that used very strong mixing or regularity assumptions to obtain distribution results. We want to use new techniques that should allow us to overcome the difficulties that are entailed by weakening the mixing properties.Let us note that research in this area is of interest to a wide variety of scientists including experimentalists who want to do numerical simulations. Detailed knowledge about the distribution of return times can be used to develop more reliable ways to numerically analyze time series of chaotic dynamical systems. The proposed research can also serve to develop more efficient data compression algorithms.