Two versions of a specific natural extension

TL;DR

This paper presents two variants of the natural extension for a particular greedy beta-transformation with deleted digits, providing explicit formulas for their invariant measures which are equivalent to Lebesgue measure.

Contribution

It introduces two versions of the natural extension for a specific beta-transformation and derives explicit invariant measures for them.

Findings

Explicit invariant measures obtained for both versions.

The measures are shown to be equivalent to Lebesgue measure.

Provides a foundation for analyzing similar transformations.

Abstract

We give two versions of the natural extension of a specific greedy beta-transformation with deleted digits. We use the natural extension to obtain an explicit expression for the invariant measure, equivalent to the Lebesgue measure, of this beta-transformation.