Recurrence and Mixing in Dynamical Systems

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The project is to research statistical and mixing properties of dynamical systems. The proposed work falls into three areas as follows:(i) (jointly with S Vaienti) The first area of research is to investigate under what conditions `cylinder neighbourhoods' if a dynamical system have mesures that are lognormally distributed. We are moreover interested in the speed of convergence to the Central Limit Theorem for a wide class of dynamical systems where the mixing is allowed to be `delayed' according to the smallness of the return neighbourhood.We expect that our techniques will also enable us to do a fine analysis of the repeat time statistics in terms of the Law of the Iterated Logarithm and the Invariance Principle.(ii) (jointly with S Vaienti) In the last three years we have developed a general scheme that can be used to determine the limiting distribution of return times. We wish to carry this scheme further by basing it on the `tower construction' which was used by L S Young to obtain rates of decay for the correlation function.(iii) (jointly with H Hu) The third area of research is on parabolic maps that have very pronounced parabolicity so that the generic invariant measure is infinite. The transfer operator in this case converges to a constant which is entirely determined by the behaviour of the map at the parabolic point. We want to determine the rate of convergence using the Hilbert metric method and some results from renewal theory.Out of the three problems considered the first two in particular have their origin in Shannon's work in information theory: Dynamical systems are characteristically analysed by using the coding which is derived by tracing orbits and identifying them according to the partition elements they pass through. In this way a `cylinder set' represents a given sequence of symbols and it is a classical problem in coding theory to determine the return statistics of this sequence. Shannon and others have shown that expected return time is typically exponential where the rate is equal to the metric entropy (i.e.\ the average information content). The study of return times has for instance been beneficial in the implementation of data compression schemes.Generally a detailed knowledge of the distribution of return times is necessary to do a reliable analysis of time series that might come from experimental data or numerical simulations. The proposed research focuses on finding general conditions for a large variety of dynamical systems under which the return times are in the limit lognormally distributed and also to provide error terms. The error terms will depend on the dynamical parameters of the system.