Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-26T17:28:33.408Z Has data issue: false hasContentIssue false
  • English
  • Français

Characteristic measures of symbolic dynamical systems

Part of: Topological dynamics

Published online by Cambridge University Press:  18 March 2021

JOSHUA FRISCH  and
OMER TAMUZ Open the ORCID record for OMER TAMUZ [Opens in a new window]
JOSHUA FRISCH
Affiliation:
California Institute of Technology, Pasadena, CA, USA (e-mail: [email protected])
OMER TAMUZ*
Affiliation:
California Institute of Technology, Pasadena, CA, USA (e-mail: [email protected])
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A probability measure is a characteristic measure of a topological dynamical system if it is invariant to the automorphism group of the system. We show that zero entropy shifts always admit characteristic measures. We use similar techniques to show that automorphism groups of minimal zero entropy shifts are sofic.

Keywords

symbolic dynamicsautomorphism groupsgroup actions

MSC classification

Primary: 37B10: Symbolic dynamics
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 5 , May 2022 , pp. 1655 - 1661
Check for updates
Copyright
© The Author(s), 2021. 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

Boyle, M., Lind, D. and Rudolph, D.. The automorphism group of a shift of finite type. Trans. Amer. Math. Soc. 306(1) (1988), 71–114.10.1090/S0002-9947-1988-0927684-2CrossRefGoogle Scholar
Ceccherini-Silberstein, T. and Coornaert, M.. Cellular Automata and Groups. Springer Science & Business Media, Heidelberg, 2010.10.1007/978-3-642-14034-1CrossRefGoogle Scholar
Coven, E. M., Quas, A. and Yassawi, R.. Computing automorphism groups of shifts using atypical equivalence classes. Discrete Anal. 3 (2016), 28 pp.Google Scholar
Cyr, V. and Kra, B.. The automorphism group of a shift of linear growth: beyond transitivity. Forum Math. Sigma 3 (2015), Paper No. e5, 27 pp.10.1017/fms.2015.3CrossRefGoogle Scholar
Cyr, V. and Kra, B.. The automorphism group of a minimal shift of stretched exponential growth. J. Mod. Dyn. 10 (2016), 483–495.Google Scholar
Cyr, V. and Kra, B.. The automorphism group of a shift of subquadratic growth. Proc. Amer. Math. Soc. 144(2) (2016), 613–621.10.1090/proc12719CrossRefGoogle Scholar
Donoso, S., Durand, F., Maass, A. and Petite, S.. On automorphism groups of low complexity subshifts. Ergod. Th. & Dynam. Sys. 36(1) (2016), 64–95.10.1017/etds.2015.70CrossRefGoogle Scholar
Furstenberg, H.. A Poisson formula for semi-simple Lie groups. Ann. of Math. (2) 77 (1963), 335–386.10.2307/1970220CrossRefGoogle Scholar
Glasner, S.. Proximal Flows (Lecture Notes in Mathematics, 517). Springer, Berlin, 1976.Google Scholar
Gromov, M.. Endomorphisms of symbolic algebraic varieties. J. Eur. Math. Soc. 1(2) (1999), 109–197.10.1007/PL00011162CrossRefGoogle Scholar
Hedlund, G. A.. Endomorphisms and automorphisms of the shift dynamical system. Math. Syst. Theory 3(4) (1969), 320–375.10.1007/BF01691062CrossRefGoogle Scholar
Juschenko, K.. Lecture Notes on Sofic Groups, 2021, https://web.ma.utexas.edu/users/juschenko/files/soficgroups.pdf.Google Scholar
Salo, V.. Toeplitz subshift whose automorphism group is not finitely generated. Colloq. Math. 146(1) (2017), 53–76.10.4064/cm6463-2-2016CrossRefGoogle Scholar
Salo, V. and Törmä, I.. Block maps between primitive uniform and Pisot substitutions. Ergod. Th. & Dynam. Sys. 35(7) (2015), 2292–2310.10.1017/etds.2014.29CrossRefGoogle Scholar
Weiss, B.. Sofic groups and dynamical systems. Sankhya A 62(3) (2000), 350–359.Google Scholar