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Ergodic optimization for continuous functions on non-Markov shifts

Part of: Topological dynamics Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  15 August 2025

MAO SHINODA ,
HIROKI TAKAHASI  and
KENICHIRO YAMAMOTO
MAO SHINODA
Affiliation:
Department of Mathematics, Ochanomizu University https://ror.org/03599d813 , 2-1-1 Otsuka, Bunkyo-ku, Tokyo 112-8610, Japan (e-mail: shinoda.mao@ocha.ac.jp)
HIROKI TAKAHASI*
Affiliation:
Department of Mathematics, Keio University https://ror.org/02kn6nx58 , Yokohama 223-8522, Japan
KENICHIRO YAMAMOTO
Affiliation:
Department of General Education, Nagaoka University of Technology https://ror.org/00ys1hz88 , Nagaoka 940-2188, Japan (e-mail: k_yamamoto@vos.nagaokaut.ac.jp)
*
e-mail: hiroki@math.keio.ac.jp
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Abstract

Ergodic optimization aims to describe dynamically invariant probability measures that maximize the integral of a given function. For a wide class of intrinsically ergodic subshifts over a finite alphabet, we show that the space of continuous functions on the shift space contains two disjoint subsets: one is a dense $G_\delta $ set for which all maximizing measures have ‘relatively small’ entropy; the other is the set of functions having uncountably many, fully supported ergodic maximizing measures with ‘relatively large’ entropy. This result generalizes and unifies the results of Morris [Discrete Contin. Dyn. Syst. 27 (2010), 383–388] and Shinoda [Nonlinearity 31 (2018), 2192–2200] on symbolic dynamics, and applies to a wide class of intrinsically ergodic non-Markov symbolic dynamics without the Bowen specification property, including any transitive piecewise monotonic interval map, some coded shifts, and multidimensional $\beta $-transformations. Along with these examples of application, we provide an example of an intrinsically ergodic subshift with positive obstruction entropy to specification.

Keywords

ergodic optimizationsubshiftthermodynamic formalismspecification

MSC classification

Primary: 37B10: Symbolic dynamics
Secondary: 37D35: Thermodynamic formalism, variational principles, equilibrium states

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , First View , pp. 1 - 28
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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