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Specification and Towers in Shift Spaces

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Abstract

We show that a shift space on a finite alphabet with a non-uniform specification property can be modeled by a strongly positive recurrent countable-state Markov shift to which every equilibrium state lifts. In addition to uniqueness of the equilibrium state, this gives strong statistical properties including the Bernoulli property, exponential decay of correlations, central limit theorem, and analyticity of pressure, which are new even for uniform specification. We give applications to shifts of quasi-finite type, synchronised and coded shifts, and factors of \({\beta}\)-shifts and S-gap shifts.

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Acknowledgement

I am grateful to the anonymous referees for many comments that improved the exposition and for spotting errors in earlier versions of the result on factors and of Lemma 7.5; the latter, which was also pointed out to me by Qu Congcong, necessitated a change in the formulation of condition [I] from previous versions. I am also grateful to Omri Sarig for clarifying aspects of strong positive recurrence as they appear in Sect. 2.1, and to Dominik Kwietniak for introducing me to cocyclic subshifts and [Kwa00, Kwa04]. This work was partially supported by NSF grants DMS-1362838 and DMS-1554794.

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  1. Department of Mathematics, University of Houston, Houston, TX, 77204, USA

    Vaughn Climenhaga

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  1. Vaughn Climenhaga
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Correspondence to Vaughn Climenhaga.

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Communicated by C. Liverani

The author is partially supported by NSF Grants DMS-1362838 and DMS-1554794.

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Climenhaga, V. Specification and Towers in Shift Spaces. Commun. Math. Phys. 364, 441–504 (2018). https://doi.org/10.1007/s00220-018-3265-y

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