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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 formulasusually involving either the velocity at which dynamical balls decreaseor by first return times for shrinking targets.In this talk we will give an overview of the concept of entropy indynamical systems and study the velocity of convergence toentropy 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 uniformlyand non-uniformly expanding shifts. Morever, we provide a topologicalcharacterization of large deviations bounds for Gibbs measures anddeduce topological aspects: the local entropy is zero and thetopological pressure of positive measure sets is total(joint work with Y. Zhao - Soochow University).
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