Basic thermodynamical formalism for sandwich subshifts

TL;DR

This paper develops a thermodynamical formalism for hereditary and subordinate subshifts, introducing measure-theoretic and two-sided analogues, with applications to $ ext{B}$-free systems, to analyze their measure-theoretic properties.

Contribution

It introduces and studies measure-theoretic and two-sided versions of hereditary and subordinate subshifts, expanding the thermodynamical formalism framework for these classes.

Findings

Characterization of measure-theoretic properties of subordinate subshifts.

Introduction of sandwich hereditary and subordinate subshifts.

Application to $ ext{B}$-free systems.

Abstract

Consider a partial order on $\{0,1\}^{\mathbb Z}: x\leq y$ when $x_i\leq y_i$ for all $i\in\mathbb{Z}$. A subshift $X\subset\{0,1\}^{\mathbb{Z}}$ is hereditary if together with any $x\in \{0,1\}^{\mathbb Z}$ it contains all $y\leq x$. Heuristically speaking, a hereditary subshift contains all the elements between maximal elements (with respect to this partial order) and the element $0^{\mathbb Z}$. In a particular situation when it suffices to take (the orbit closure of) all the elements between a single maximal element $x$ and the element $0^{\mathbb Z}$, we speak of subordinate subshifts. In this paper we investigate measure-theoretic properties of such subshifts, with a special emphasis on thermodynamical formalism. The key notion is a measure-theoretic counterpart of subordinate subshifts, where the role of a single maximal element is replaced with a single (maximal with respect to a certain order) invariant measure on $\{0,1\}^{\mathbb{Z}}$. We also introduce and investigate two-sided analogues of the above classes, we call them {\it sandwich hereditary}, {\it sandwich subordinate} and {\it sandwich measure-theoretically subordinate} subshifts. Sandwich hereditary subshifts can be thought of as sets of elements between some pairs of maximal and minimal elements satisfying certain assumptions. Sandwich subordinate subshifts occur when it suffices to take (the orbit closure of) all the elements between a single pair of sequences $(w,x)$, where $w\leq x$. In sandwich measure-theoretically subordinate subshifts, the role of a pair of sequences is replaced by a pair (precisely speaking: a joining) of two invariant measures on $\{0,1\}^{\mathbb{Z}}$. The notions and results are motivated by those from the theory of so-called $\mathscr{B}$-free systems.