Statistical Behaviour of Recurrence in Dynamical Systems

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Abstract:The work proposed is to study the connection between mixing properties of dynamical systems and their statistical properties. The goal is to develop a general approach that will enable us to obtain results on the distribution of normalised return times in a wide variety of settings including non-uniformly hyperbolic dynamical systems. This extends to the return time distributions for the canonical invariant measure interval maps with parabolic points or for equilibrium states for rational maps which have critical points in the Julia set. We expect to obtain explicit error terms for the distribution of higher order returns of any order. Ultimately we hope to extend our approach to L-S Young's `tower construction' which originally was introduced to obtain rates on the decay of correlations.This area of research is of interest to a wide variety of scientist. For instance experimentalists who want to do numerical simulations will benefit from the proposed problems which will assist them in developing more reliable ways to numerically analyse time series of chaotic dynamical systems. The proposed research can also serve to develop more efficient data compression algorithms since it quantifies the error in the number of times with which a newly coded sequence of symbols occurs from its predicted average value.