New limit properties for infinite measure preserving systems

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A dynamical system is a mathematical model describing the time evolution by differential equations or iterated mappings. A stochastic process is a mathematical description of a sequence of chance events. Apart from simple mathematical models, very little is known about (strong) mixing for dynamical systems preserving an infinite measure. Roughly speaking, 'mixing' measures how rapidly initial information is lost and in a sense, indicates how much the dynamical system differs from an independent stochastic process. The first part of the proposed project focuses on proving stochastic properties and specifically, strong mixing of continuous time dynamical systems that can be represented as group extensions of semiflows and perturbed versions of these. These systems are inspired by physical models of Lorentz gas, i.e., particles bouncing in a pattern of scatterers. This area has seen a surge of recent activity, with some important breakthroughs in the last decade, but for perturbed systems, which correspond to non-periodic patterns of scatterers, the behaviour is still a wide open question. These systems have been studied with a variety of methods, but the development of operator renewal theory provides new inroads. As the relevant groups are non-compact and require infinite measures, one strength we see in developing/applying operator renewal theory for such flows is the potential to address cases where the corresponding distributions have heavier tails than could be treated before. More importantly, the proposed research addresses the possibility to go beyond group extension, by studying their perturbed versions.Previous results on strong mixing for infinite measure preserving systems are obtained when the condition of regularly varying tails of certain return time distributions is satisfied. A second direction of my project revolves around the following question: does strong mixing make sense in the absence of regular variation? Within this topic, we aim to formulate and prove a version of (strong) mixing along subsequences and such an appropriate version of Wiener's lemma along subsequences. One of the (longer term) aims here of the second part of the proposal is to provide an analytic proof (the existing proof uses probabilistic methods) of the Erdos Feller Pollard Theorem for renewal sequences with infinite mean (not necessarily, regularly varying).