A natural extension for the greedy beta-transformation with three deleted digits

TL;DR

This paper derives an explicit invariant measure for a specific greedy beta-transformation with three deleted digits, establishing its exactness and weak Bernoulli property through a natural extension.

Contribution

It provides the first explicit formula for the invariant measure of this class of beta-transformations with deleted digits, extending previous results.

Findings

Explicit invariant measure derived

Transformation shown to be exact

Transformation proven to be weakly Bernoulli

Abstract

We give an explicit expression for the invariant measure, absolutely continuous with respect to the Lebesgue measure, of the greedy beta-transformation with three deleted digits. We define a version of the natural extension of the transformation to obtain this expression. We get that the transformation is exact and weakly Bernoulli.