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Subsystems of transitive subshifts with linear complexity

Part of: Topological dynamics Discrete mathematics in relation to computer science Ergodic theory

Published online by Cambridge University Press:  17 March 2021

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

Keywords

generic measureslinear complexitysymbolic dynamics

MSC classification

Primary: 37B10: Symbolic dynamics
Secondary: 37A25: Ergodicity, mixing, rates of mixing 68R15: Combinatorics on words
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 6 , June 2022 , pp. 1967 - 1993
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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

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