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Symbolic dynamics for non-uniformly hyperbolic systems

Part of: Dynamical systems with hyperbolic behavior Dynamical systems and ergodic theory Topological dynamics Smooth dynamical systems: general theory

Published online by Cambridge University Press:  17 December 2020

YURI LIMA
YURI LIMA*
Affiliation:
Yuri Lima, Departamento de Matemática, Universidade Federal do Ceará (UFC), Campus do Pici, Bloco 914, CEP 60440-900. Fortaleza – CE, Brazil (e-mail: [email protected])
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Abstract

This survey describes the recent advances in the construction of Markov partitions for non-uniformly hyperbolic systems. One important feature of this development comes from a finer theory of non-uniformly hyperbolic systems, which we also describe. The Markov partition defines a symbolic extension that is finite-to-one and onto a non-uniformly hyperbolic locus, and this provides dynamical and statistical consequences such as estimates on the number of closed orbits and properties of equilibrium measures. The class of systems includes diffeomorphisms, flows, and maps with singularities.

Keywords

Markov partitionPesin theorysymbolic dynamics

MSC classification

Primary: 37-02: Research exposition (monographs, survey articles) 37B10: Symbolic dynamics 37C05: Smooth mappings and diffeomorphisms 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
Secondary: 37C35: Orbit growth 37D35: Thermodynamic formalism, variational principles, equilibrium states
Type
Survey Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 9 , September 2021 , pp. 2591 - 2658
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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