Asymptotic -Algebras from -Actions on Higher Rank Graphs

Inhyeop Yi

doi:10.1155/2013/752497

Abstract

For a dynamical system arising from -action on a higher rank graph with finite vertex set, we show that the semidirect product of the asymptotic equivalence relation groupoid is essentially principal if and only if the -graph satisfies the aperiodic condition. Then we show that the corresponding asymptotic Ruelle algebra is simple if the graph is primitive with the aperiodic condition.

Highlights

In [1], Kumjian and Pask constructed Zk-action dynamical systems on higher rank graphs, which are higher dimensional analog of subshifts of finite type in symbolic dynamics.Naturally their dynamical systems exhibit many of the same dynamical properties in subshifts of finite type and Smale spaces studied by Putnam [2], Putnam and Spielberg [3], and Ruelle [4, 5]
For a dynamical system arising from Zk-action on a higher rank graph with finite vertex set, we show that the semidirect product of the asymptotic equivalence relation groupoid is essentially principal if and only if the k-graph satisfies the aperiodic condition
Their dynamical systems exhibit many of the same dynamical properties in subshifts of finite type and Smale spaces studied by Putnam [2], Putnam and Spielberg [3], and Ruelle [4, 5]

Summary

Introduction

In [1], Kumjian and Pask constructed Zk-action dynamical systems on higher rank graphs, which are higher dimensional analog of subshifts of finite type in symbolic dynamics.Naturally their dynamical systems exhibit many of the same dynamical properties in subshifts of finite type and Smale spaces studied by Putnam [2], Putnam and Spielberg [3], and Ruelle [4, 5]. For a dynamical system arising from Zk-action on a higher rank graph with finite vertex set, we show that the semidirect product of the asymptotic equivalence relation groupoid is essentially principal if and only if the k-graph satisfies the aperiodic condition.

Results

Conclusion