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Speed of convergence to entropy: probabilistic, topological
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The notion of measure theoretical entropy, brought from the Statistical Mechanics, is one of the measurements of complexity in dynamical systems.
Due to the importance of this concept there are several limit formulas
usually involving either the velocity at which dynamical balls decrease
or by first return times for shrinking targets.
In this talk we will give an overview of the concept of entropy in
dynamical systems and study the velocity of convergence to
entropy in the Shannon-McMillan-Breiman convergence.
In fact, we have (i) exponential large deviation bounds for weak Gibbs measures and topologically mixing subshifts of finite type; (ii) almost sure estimates for the error in the approximation of entropy for uniformly
and non-uniformly expanding shifts. Morever, we provide a topological
characterization of large deviations bounds for Gibbs measures and
deduce topological aspects: the local entropy is zero and the
topological pressure of positive measure sets is total
(joint work with Y. Zhao - Soochow University).
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