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Decay of correlations for weakly expanding dynamical systems with Dini potentials under optimal quasi-gap condition. (English) Zbl 1493.37023

The paper contributes to thermodynamic formalism in ergodic theory. It extends the classical results on spectral gaps for the Ruelle operator to a class of weakly expanding systems and Dini potentials which may not have spectral gaps, but still some of the consequences of the theory apply. An historical and interesting overview about the development of the research in this area is given in the introduction.
The main result in the paper is Theorem 2.5 which applies to weakly expanding systems (defined at the beginning of Section 2 and generalizing classical expanding maps with Markov partitions) and potentials satisfying some technical conditions stated in Definition 2.3.

MSC:

37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
37D35 Thermodynamic formalism, variational principles, equilibrium states for dynamical systems
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