Isomorphisms between positive and negative beta-transformations

TL;DR

This paper investigates the relationship between positive and negative beta-transformations, establishing conditions under which they are measurably isomorphic, particularly for multinacci numbers, and demonstrating non-isomorphism for other beta values.

Contribution

It proves the existence of measurable isomorphisms between positive and negative beta-transformations for multinacci numbers and shows non-isomorphism for other beta values between 1 and 2.

Findings

Measurable isomorphisms exist for multinacci beta values.

No isomorphism for beta values between 1 and 2 outside multinacci numbers.

Characterization of beta values where transformations are isomorphic.

Abstract

We compare a piecewise linear map with constant slope beta>1 and a piecewise linear map with constant slope -beta. These maps are called the positive and negative beta-transformations. We show that for a certain set of beta's, the multinacci numbers, there exists a measurable isomorphism between these two maps. We further show that for for all other values of beta between 1 and 2 the two maps cannot be isomorphic.