Specification and towers in shift spaces

TL;DR

This paper demonstrates that shift spaces with a non-uniform specification property can be modeled by a specific Markov shift, leading to strong statistical properties and broad applications in symbolic dynamics.

Contribution

It introduces a modeling approach for non-uniform specification shift spaces using strongly positive recurrent Markov shifts, establishing new statistical properties and applications.

Findings

Unique equilibrium states with strong statistical properties

Exponential decay of correlations and Bernoulli property

Applicability to various classes of shifts like beta-shifts and coded shifts

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.