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Subsystems of transitive subshifts with linear complexity

Published online by Cambridge University Press:  17 March 2021

ANDREW DYKSTRA
Affiliation:
Department of Mathematics, Hamilton College, Clinton, NY13323, USA (e-mail: [email protected])
NICHOLAS ORMES
Affiliation:
Department of Mathematics, University of Denver, 2390 S. York St., Denver, CO80208, USA (e-mail: [email protected]), url: www.math.du.edu/∼rpavlov/
RONNIE PAVLOV*
Affiliation:
Department of Mathematics, University of Denver, 2390 S. York St., Denver, CO80208, USA (e-mail: [email protected]), url: www.math.du.edu/∼rpavlov/
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Abstract

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We bound the number of distinct minimal subsystems of a given transitive subshift of linear complexity, continuing work of Ormes and Pavlov [On the complexity function for sequences which are not uniformly recurrent. Dynamical Systems and Random Processes (Contemporary Mathematics, 736). American Mathematical Society, Providence, RI, 2019, pp. 125--137]. We also bound the number of generic measures such a subshift can support based on its complexity function. Our measure-theoretic bounds generalize those of Boshernitzan [A unique ergodicity of minimal symbolic flows with linear block growth. J. Anal. Math.44(1) (1984), 77–96] and are closely related to those of Cyr and Kra [Counting generic measures for a subshift of linear growth. J. Eur. Math. Soc.21(2) (2019), 355–380].

Type
Original Article
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

References

Birkhoff, G. D.. Quelques théorèmes sur le mouvement des systèmes dynamiques. Bull. Soc. Math. France 40 (1912), 305323.CrossRefGoogle Scholar
Boshernitzan, M.. A unique ergodicity of minimal symbolic flows with linear block growth. J. Anal. Math. 44(1) (1984), 7796.CrossRefGoogle Scholar
Cyr, V. and Kra, B.. Counting generic measures for a subshift of linear growth. J. Eur. Math. Soc. 21(2) (2019), 355380.CrossRefGoogle Scholar
Damron, M. and Fickenscher, J.. On the number of ergodic measures for minimal shifts with eventually constant complexity growth. Ergod. Th. & Dynam. Sys. 37(7) (2017), 20992130.CrossRefGoogle Scholar
Damron, M. and Fickenscher, J.. The number of ergodic measures for transitive subshifts under the regular bispecial condition. Preprint.Google Scholar
Katok, A.. Invariant measures of flows on orientable surfaces. Dokl. Akad. Nauk 211 (1973), 775778.Google Scholar
Kůrka, P.. Topological and Symbolic Dynamics. Société Mathématique de France, Paris, 2003.Google Scholar
Lind, D. and Marcus, B.. An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
Morse, M. and Hedlund, G. A.. Symbolic dynamics II. Sturmian trajectories. Amer. J. Math. 62 (1940), 142.CrossRefGoogle Scholar
Ormes, N. and Pavlov, R.. On the complexity function for sequences which are not uniformly recurrent. Dynamical Systems and Random Processes (Contemporary Mathematics, 736). American Mathematical Society, Providence, RI, 2019, pp.125137.CrossRefGoogle Scholar
Paul, M. E.. Minimal symbolic flows having minimal block growth. Math. Syst. Theory 8(4) (1974–1975), 309315.CrossRefGoogle Scholar
Fogg, N. Pytheas. Substitutions in Dynamics, Arithmetics and Combinatorics (Lecture Notes in Mathematics, 1794). Springer, Berlin, 2002.CrossRefGoogle Scholar
Veech, W.. Interval exchange transformations. J. Anal. Math. 33 (1978), 222272.CrossRefGoogle Scholar
Walters, P.. An Introduction to Ergodic Theory. Springer, New York, 1982.CrossRefGoogle Scholar