Large deviation principle for linear mod 1 transformations

TL;DR

This paper establishes a large deviation principle for a class of linear mod 1 transformations, using periodic measures and Markov diagrams, contributing to the understanding of their statistical properties.

Contribution

It proves the level-2 large deviation principle for linear mod 1 transformations with specific parameters, a novel result in dynamical systems.

Findings

Validates the large deviation principle for the specified transformations

Uses density of periodic measures and Hofbauer's Markov Diagram in proof

Identifies the measure of maximal entropy as unique equilibrium measure

Abstract

For $0\le \alpha <1$ and $\beta>2$, we consider a linear mod 1 transformation on a unit interval; $x\mapsto\beta x+\alpha$ (${\rm mod}\ 1$), and prove that it satisfies the level-2 large deviation principle with the unique measure of maximal entropy. For the proof, we use the density of periodic measures and Hofbauer's Markov Diagram.