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$\overline {d}$-continuity for countable state shifts

Part of: Topological dynamics Dynamical systems with hyperbolic behavior Ergodic theory

Published online by Cambridge University Press:  03 October 2025

JASMINE BHULLAR
JASMINE BHULLAR*
Affiliation:
Department of Mathematics, Tufts University , Medford, MA 02155, USA
*
e-mail: jbhull01@tufts.edu
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Abstract

For full shifts on finite alphabets, Coelho and Quas [Criteria for $\overline {d}$-continuity. Trans. Amer. Math. Soc. 350(8) (1998), 3257–3268] showed that the map that sends a Hölder continuous potential $\phi $ to its equilibrium state $\mu _\phi $ is $\overline {d}$-continuous. We extend this result to the setting of full shifts on countable (infinite) alphabets. As part of the proof, we show that the map that sends a strongly positive recurrent potential to its normalization is continuous for potentials on mixing countable state Markov shifts.

Keywords

classical ergodic theorysymbolic dynamics

MSC classification

Primary: 37D35: Thermodynamic formalism, variational principles, equilibrium states
Secondary: 37B10: Symbolic dynamics 37A30: Ergodic theorems, spectral theory, Markov operators

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , First View , pp. 1 - 24
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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