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On subshifts with slow forbidden word growth

Part of: Topological dynamics Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  12 January 2021

RONNIE PAVLOV
RONNIE PAVLOV*
Affiliation:
Department of Mathematics, University of Denver, 2390 S. York St., Denver, CO80208, USA (www.math.du.edu/~rpavlov/)
*
e-mail: rpavlov@du.edu
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Abstract

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In this work, we treat subshifts, defined in terms of an alphabet $\mathcal {A}$ and (usually infinite) forbidden list $\mathcal {F}$ , where the number of n-letter words in $\mathcal {F}$ has ‘slow growth rate’ in n. We show that such subshifts are well behaved in several ways; for instance, they are boundedly supermultiplicative in the sense of Baker and Ghenciu [Dynamical properties of S-gap shifts and other shift spaces. J. Math. Anal. Appl.430(2) (2015), 633–647] and they have unique measures of maximal entropy with the K-property and which satisfy Gibbs bounds on large (measure-theoretically) sets. The main tool in our proofs is a more general result, which states that bounded supermultiplicativity and a sort of measure-theoretic specification property together imply uniqueness of the measure of maximum entropy and our Gibbs bounds. We also show that some well-known classes of subshifts can be treated by our results, including the symbolic codings of $x \mapsto \alpha + \beta x$ (the so-called $\alpha $ - $\beta $ shifts of Hofbauer [Maximal measures for simple piecewise monotonic transformations. Z. Wahrsch. verw. Geb.52(3) (1980), 289–300]) and the bounded density subshifts of Stanley [Bounded density shifts. Ergod. Th. & Dynam. Sys.33(6) (2013), 1891–1928].

Keywords

Gibbs boundsintrinsic ergodicitysymbolic dynamics

MSC classification

Primary: 37B10: Symbolic dynamics
Secondary: 37B40: Topological entropy 37D35: Thermodynamic formalism, variational principles, equilibrium states
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 4 , April 2022 , pp. 1487 - 1516
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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