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Symbolic dynamics for pointwise hyperbolic systems on open regions

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

Published online by Cambridge University Press:  10 September 2024

CHUPENG WU  and
YUNHUA ZHOU
CHUPENG WU
Affiliation:
College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China (e-mail: 20200601008@cqu.edu.cn)
YUNHUA ZHOU*
Affiliation:
College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China (e-mail: 20200601008@cqu.edu.cn)
*
e-mail: zhouyh@cqu.edu.cn
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Abstract

Under certain conditions, we construct a countable Markov partition for pointwise hyperbolic diffeomorphism $f:M\rightarrow M$ on an open invariant subset $O\subset M$, which allows the Lyapunov exponents to be zero. From this partition, we define a symbolic extension that is finite-to-one and onto a subset of O that carries the same finite f-invariant measures as O. Our method relies upon shadowing theory of a recurrent-pointwise-pseudo-orbit that we introduce. As a canonical application, we estimate the number of closed orbits for f.

Keywords

pointwise hyperbolicityMarkov partitionsymbolic dynamics

MSC classification

Primary: 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) 37C05: Smooth mappings and diffeomorphisms
Secondary: 37B10: Symbolic dynamics 37C35: Orbit growth

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 45 , Issue 2 , February 2025 , pp. 595 - 648
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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