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Balanced strong shift equivalence, balanced in-splits, and eventual conjugacy

Part of: Ergodic theory Topological dynamics

Published online by Cambridge University Press:  04 December 2020

KEVIN AGUYAR BRIX Open the ORCID record for KEVIN AGUYAR BRIX [Opens in a new window]
KEVIN AGUYAR BRIX*
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW2522, Australia
*
e-mail:kabrix.math@fastmail.com
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Abstract

We introduce the notion of balanced strong shift equivalence between square non-negative integer matrices, and show that two finite graphs with no sinks are one-sided eventually conjugate if and only if their adjacency matrices are conjugate to balanced strong shift equivalent matrices. Moreover, we show that such graphs are eventually conjugate if and only if one can be reached by the other via a sequence of out-splits and balanced in-splits, the latter move being a variation of the classical in-split move introduced by Williams in his study of shifts of finite type. We also relate one-sided eventual conjugacies to certain block maps on the finite paths of the graphs. These characterizations emphasize that eventual conjugacy is the one-sided analog of two-sided conjugacy.

Keywords

directed graphseventual conjugacybalanced strong shift equivalencemoves on graphs

MSC classification

Primary: 37B10: Symbolic dynamics
Secondary: 37A55: Relations with the theory of $C^*$-algebras
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 1 , January 2022 , pp. 19 - 39
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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