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Weighted topological pressure revisited

Part of: Dynamical systems with hyperbolic behavior Topological dynamics Smooth dynamical systems: general theory Classical measure theory Ergodic theory Measure-theoretic ergodic theory

Published online by Cambridge University Press:  08 May 2024

NIMA ALIBABAEI Open the ORCID record for NIMA ALIBABAEI [Opens in a new window]
NIMA ALIBABAEI*
Affiliation:
Department of Mathematics, Kyoto University, Kyoto 606-8501, Japan
*
e-mail: alibabaei.nima.28c@st.kyoto-u.ac.jp
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Abstract

Feng and Huang [Variational principle for weighted topological pressure. J. Math. Pures Appl. (9) 106 (2016), 411–452] introduced weighted topological entropy and pressure for factor maps between dynamical systems and established its variational principle. Tsukamoto [New approach to weighted topological entropy and pressure. Ergod. Th. & Dynam. Sys. 43 (2023), 1004–1034] redefined those invariants quite differently for the simplest case and showed via the variational principle that the two definitions coincide. We generalize Tsukamoto’s approach, redefine the weighted topological entropy and pressure for higher dimensions, and prove the variational principle. Our result allows for an elementary calculation of the Hausdorff dimension of affine-invariant sets such as self-affine sponges and certain sofic sets that reside in Euclidean space of arbitrary dimension.

Keywords

dynamical systemsweighted topological entropyweighted topological pressurevariational principleaffine-invariant setsself-affine spongessofic setsHausdorff dimension

MSC classification

Primary: 28A80: Fractals 28D20: Entropy and other invariants 37D35: Thermodynamic formalism, variational principles, equilibrium states
Secondary: 37B40: Topological entropy 37A35: Entropy and other invariants, isomorphism, classification 37C45: Dimension theory of dynamical systems
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , First View , pp. 1 - 37
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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