Continuous Orbit Equivalence on Self-Similar Graph Actions

Inhyeop Yi

doi:10.3390/math7100990

Abstract

For self-similar graph actions, we show that isomorphic inverse semigroups associated to a self-similar graph action are a complete invariant for the continuous orbit equivalence of inverse semigroup actions on infinite path spaces.

Highlights

Following the groundbreaking result of Giordano, Putnam, and Skau concerning orbit equivalence on Cantor minimal systems [1], Matsumoto [2] introduced the continuous orbit equivalence of one-sided subshifts of finite type
Li [3] showed that the continuous orbit equivalence of graphs is equivalent to the continuous orbit equivalence of the actions of groups generated by the edge sets of graphs to the infinite path spaces of graphs
Cordeiro and Beuter [4] showed that the continuous orbit equivalence of graphs is equivalent to the continuous orbit equivalence of the actions of inverse semigroups that are naturally associated with graphs on the infinite path spaces of graphs

Summary

Introduction

Following the groundbreaking result of Giordano, Putnam, and Skau concerning orbit equivalence on Cantor minimal systems [1], Matsumoto [2] introduced the continuous orbit equivalence of one-sided subshifts of finite type. Cordeiro and Beuter [4] showed that the continuous orbit equivalence of graphs is equivalent to the continuous orbit equivalence of the actions of inverse semigroups that are naturally associated with graphs on the infinite path spaces of graphs. Combining these two results for graphs with mild restrictions, it follows that the continuous orbit equivalence of group actions is equivalent to that of inverse semigroup actions. We discuss some reasons behind this difficulty and directions for future steps

Self-Similar Graph Actions

Continuous Orbit Equivalences

A Remark on GROUP actions