From rates of mixing to recurrence times via large deviations

By José F. Alves, J. Freitas, Stefano Luzzatto, Sandro Vaienti. 2010

82 citations recorded by the source corpus.

Abstract A classic approach in dynamical systems is to use particular geometric structures to deduce statistical properties, for example the existence of invariant measures with stochastic-like behaviour such as large deviations or decay of correlations. Such geometric structures are generally highly non-trivial and thus a natural question is the extent to which this approach can be applied. In this paper we show that in many cases stochastic-like behaviour itself implies that the system has certain non-trivial geometric properties, which are therefore necessary as well as sufficient conditions for the occurrence of the statistical properties under consideration. As a by product of our techniques we also obtain some new results on large deviations for certain classes of systems which include Viana maps and multidimensional piecewise expanding maps.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8cb26651e62d7662ae30a7ae5b824e17c6384ef7

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Almost sure invariance principle for sequential and non-stationary dynamical systems (2014, background): “We give here a multidimensional version of the maps considered in the preceding section; these maps were extensively investigated in [2,3,21,22,35] and we defer to those papers for more details.”
Martingale–coboundary decomposition for families of dynamical systems (2016, background): “Indeed, it is generally true that nonuniform expansivity plus the existence of good statistical properties actually implies the existence of an inducing scheme satisfying the conditions above, see [4, 5].”