Hostname: page-component-848d4c4894-nmvwc Total loading time: 0 Render date: 2024-07-05T12:15:08.315Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Topological dynamics Ergodic theory

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:[email protected]
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Bates, T. and Pask, D.. Flow equivalence of graph algebras. Ergod. Th. & Dynam. Sys. 24 (2004), 367–382.CrossRefGoogle Scholar
Boyle, M., Franks, J. and Kitchens, B.. Automorphisms of one-sided subshifts of finite type. Ergod. Th. & Dynam. Sys. 10(3) (1990), 421–449.CrossRefGoogle Scholar
Brix, K. A. and Carlsen, T. M.. Cuntz–Krieger algebras and one-sided conjugacy of shift of finite type and their groupoids. J. Aust. Math. Soc. doi:10.1017/S1446788719000168. Published online 21 June 2019.CrossRefGoogle Scholar
Brownlowe, N., Carlsen, T. M. and Whittaker, M.. Graph algebras and orbit equivalence. Ergod. Th. & Dynam. Sys. 37(2) (2017), 389–417.CrossRefGoogle Scholar
Carlsen, T. M., Eilers, S., Ortega, E. and Restorff, G.. Flow equivalence and orbit equivalence for shifts of finite type and isomorphism of their groupoids. J. Math. Anal. Appl. 469 (2019), 1088–1110.CrossRefGoogle Scholar
Carlsen, T. M. and Rout, J.. Diagonal-preserving gauge-invariant isomorphisms of graph ${C}^{\ast }$ -algebras. J. Funct. Anal. 273 (2017), 2981–2993.CrossRefGoogle Scholar
Cuntz, J. and Krieger, W.. A class of ${C}^{\ast }$ -algebras and topological Markov chains. Invent. Math. 56 (1980), 251–268.CrossRefGoogle Scholar
Eilers, S. and Ruiz, E.. Refined moves for structure-preserving isomorphism of graph ${\mathrm{C}}^{\ast }$ -algebras. Preprint, 2019, arXiv:1908.03714v1 [math.OA], p. 46.Google Scholar
Kim, K. H and Roush, F. W.. The Williams conjecture is false for irreducible subshifts. Ann. of Math. (2) 149 (1999), 545–558.CrossRefGoogle Scholar
Kim, K. H. and Roush, F. W.. Decidability of shift equivalence. Dynamical Systems (College Park, MD, 1986–87) (Lecture notes in Mathematics, 1342). Springer, Berlin, 1988, pp. 374–424.Google Scholar
Kitchens, B. P.. Symbolic Dynamics: One-Sided, Two-Sided and Countable State Markov Shifts (Universitext). Springer, Berlin, 1998.CrossRefGoogle Scholar
Lind, D. and Marcus, B.. An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
Matsumoto, K.. Continuous orbit equivalence, flow equivalence of Markov shifts and circle actions on Cuntz-Krieger algebras. Math. Z. 285 (2017), 121–141.CrossRefGoogle Scholar
Matsumoto, K.. Uniformly continuous orbit equivalence of Markov shifts and gauge actions on Cuntz–Krieger algebras. Proc. Amer. Math. Soc. 145 (2017), 1131–1140.CrossRefGoogle Scholar
Matsumoto, K. and Matui, H.. Continuous orbit equivalence of topological Markov shifts and Cuntz–Krieger algebras. Kyoto J. Math. 54 (2014), 863–877.CrossRefGoogle Scholar
Parry, B. and Sullivan, D.. A topological invariant of flows on $1$ -dimensional spaces. Topology 14 (1975), 297–299.CrossRefGoogle Scholar
Williams, R. F.. Classification of subshifts of finite type. Ann. of Math. (2) 98 (1973), 120–153; Erratum, Ann. of Math. (2) 99 (1974), 380–381.CrossRefGoogle Scholar