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Equilibrium measures for two-sided shift spaces via dimension theory

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

Published online by Cambridge University Press:  10 September 2024

VAUGHN CLIMENHAGA Open the ORCID record for VAUGHN CLIMENHAGA [Opens in a new window]  and
JASON DAY Open the ORCID record for JASON DAY [Opens in a new window]
VAUGHN CLIMENHAGA
Affiliation:
Department of Mathematics, University of Houston, Houston, TX 77204, USA (e-mail: climenha@math.uh.edu)
JASON DAY*
Affiliation:
Department of Mathematics, University of Houston, Houston, TX 77204, USA (e-mail: climenha@math.uh.edu)
*
e-mail: jjday@uh.edu
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Abstract

Given a two-sided shift space on a finite alphabet and a continuous potential function, we give conditions under which an equilibrium measure can be described using a construction analogous to Hausdorff measure that goes back to the work of Bowen. This construction was previously applied to smooth uniformly and partially hyperbolic systems by the first author, Pesin, and Zelerowicz. Our results here apply to all subshifts of finite type and Hölder continuous potentials, but extend beyond this setting, and we also apply them to shift spaces with synchronizing words.

Keywords

symbolic dynamicsdimension theoryequilibrium states

MSC classification

Primary: 37D35: Thermodynamic formalism, variational principles, equilibrium states
Secondary: 37B10: Symbolic dynamics 37C45: Dimension theory of dynamical systems

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 45 , Issue 2 , February 2025 , pp. 427 - 466
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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