A monotone isomorphism theorem

TL;DR

This paper proves that for Bernoulli shifts on two symbols with equal entropy, there exists a monotone isomorphism that maps sequences to coordinatewise smaller or equal sequences, demonstrating a structured form of isomorphism.

Contribution

It establishes the existence of a monotone isomorphism between Bernoulli shifts of equal entropy, providing a new structured approach to their classification.

Findings

Existence of a monotone isomorphism for Bernoulli shifts

Monotone isomorphism maps sequences to coordinatewise smaller sequences

Extends understanding of isomorphisms in symbolic dynamics

Abstract

In the simple case of a Bernoulli shift on two symbols, zero and one, by permuting the symbols, it is obvious that any two equal entropy shifts are isomorphic. We show that the isomorphism can be realized by a factor that maps a binary sequence to another that is coordinatewise smaller than or equal to the original sequence.