On the one-way specification property and large deviations for non-transitive systems

TL;DR

This paper introduces the one-way specification property for non-transitive systems, demonstrates its applicability through examples, and establishes a large deviation principle for the $(-\beta)$-transformation under specific conditions.

Contribution

It defines a new weaker form of the specification property and applies it to prove large deviation principles for non-transitive systems like the $(-\beta)$-transformation.

Findings

The $(-\beta)$-transformation satisfies a level-2 large deviation principle.

The rate function for the large deviations is the free energy.

The property holds when $\beta>1$ is a Yrrap number.

Abstract

We introduce a weaker form of the specification property, called "one-way specification property", and give several examples of non-transitive systems satisfying this property. As an application, we show that the $(-\beta)$-transformation satisfies a level-2 large deviation principle with the Lebesgue measure and the rate function is the free energy under the condition that $\beta>1$ is a Yrrap number, that is, the orbit of $1$ under the $(-\beta)$-transformation is eventually periodic.