Density of periodic measures and large deviation principle for generalized $(\alpha, \beta)$-transformations

TL;DR

This paper introduces generalized $(ta,eta)$-transformations, proves they satisfy a large deviation principle, and establishes the density of periodic measures among ergodic measures, advancing understanding of their statistical properties.

Contribution

It extends the class of transformations known to satisfy the large deviation principle and proves the density of periodic measures in this broader context.

Findings

All transitive generalized $(ta,eta)$-transformations satisfy the level-2 large deviation principle.

Periodic measures are dense in the set of ergodic measures for these transformations.

The results unify and generalize previous work on $(ta,eta)$-transformations.

Abstract

We introduce generalized $(\alpha,\beta)$-transformations, which include all $(\alpha,\beta)$ and generalized $\beta$-transformations, and prove that all transitive generalized $(\alpha,\beta)$-transformations satisfy the level-2 large deviation principle with a unique measure of maximal entropy. A crucial step in our proof is to establish density of periodic measures in the set of ergodic measures.