Orbit equivalent substitution dynamical systems and complexity

TL;DR

This paper constructs infinitely many non-isomorphic substitution dynamical systems that are orbit equivalent, demonstrating that orbit equivalence does not imply isomorphism, and explores their complexity and Bratteli-Vershik representations.

Contribution

It provides explicit constructions of non-isomorphic systems orbit equivalent to a given substitution system and analyzes their complexity differences.

Findings

Constructed countably many non-isomorphic systems orbit equivalent to a given one.

Showed complexity differences prevent isomorphism among these systems.

Identified minimal Bratteli-Vershik models for orbit equivalence.

Abstract

For any primitive proper substitution \sigma, we give explicit constructions of countably many pairwise non-isomorphic substitution dynamical systems {(X_{\zeta_n}, T_{\zeta_n})}_{n=1}^{\infty} such that they all are (strong) orbit equivalent to (X_{\sigma}, T_{\sigma}). We show that the complexity of the substitution dynamical systems {(X_{\zeta_n}, T_{\zeta_n})} is essentially different that prevents them from being isomorphic. Given a primitive (not necessarily proper) substitution \tau, we find a stationary simple properly ordered Bratteli diagram with the least possible number of vertices such that the corresponding Bratteli-Vershik system is orbit equivalent to (X_{\tau}, T_{\tau}).