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Combinatorics of one-dimensional simple Toeplitz subshifts

Part of: Global differential geometry Discrete mathematics in relation to computer science Topological dynamics

Published online by Cambridge University Press:  13 November 2018

DANIEL SELL
DANIEL SELL*
Affiliation:
Friedrich-Schiller-Universität Jena, Institut für Mathematik, 07743 Jena, Germany email [email protected]
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Abstract

This paper provides a systematic study of fundamental combinatorial properties of one-dimensional, two-sided infinite simple Toeplitz subshifts. Explicit formulas for the complexity function, the palindrome complexity function and the repetitivity function are proved. Moreover, a complete description of the de Bruijn graphs of the subshifts is given. Finally, the Boshernitzan condition is characterized in terms of combinatorial quantities, based on a recent result of Liu and Qu [Uniform convergence of Schrödinger cocycles over simple Toeplitz subshift. Ann. Henri Poincaré12(1) (2011), 153–172]. Particular simple characterizations are provided for simple Toeplitz subshifts that correspond to the orbital Schreier graphs of the family of Grigorchuk’s groups, a class of subshifts that serves as the main example throughout the paper.

Keywords

symbolic dynamicssimple Toeplitz wordscombinatorics on words

MSC classification

Primary: 68R15: Combinatorics on words
Secondary: 37B10: Symbolic dynamics 53C23: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 6 , June 2020 , pp. 1673 - 1714
Copyright
© Cambridge University Press, 2018

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