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The number of ergodic measures for transitive subshifts under the regular bispecial condition

Part of: Ergodic theory Measure-theoretic ergodic theory Topological dynamics

Published online by Cambridge University Press:  29 December 2020

MICHAEL DAMRON  and
JON FICKENSCHER
MICHAEL DAMRON
Affiliation:
School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA30332, USA (e-mail:[email protected])
JON FICKENSCHER*
Affiliation:
Mathematics Department, Princeton University, Fine Hall, Washington Road, Princeton, NJ08544, USA
*
e-mail:[email protected]
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Abstract

If $\mathcal {A}$ is a finite set (alphabet), the shift dynamical system consists of the space $\mathcal {A}^{\mathbb {N}}$ of sequences with entries in $\mathcal {A}$ , along with the left shift operator S. Closed S-invariant subsets are called subshifts and arise naturally as encodings of other systems. In this paper, we study the number of ergodic measures for transitive subshifts under a condition (‘regular bispecial condition’) on the possible extensions of words in the associated language. Our main result shows that under this condition, the subshift can support at most $({K+1})/{2}$ ergodic measures, where K is the limiting value of $p(n+1)-p(n)$ , and p is the complexity function of the language. As a consequence, we answer a question of Boshernitzan from 1984, providing a combinatorial proof for the bound on the number of ergodic measures for interval exchange transformations.

Keywords

symbolic dynamicsclassical ergodic theoryformal languages

MSC classification

Primary: 37B10: Symbolic dynamics
Secondary: 28D05: Measure-preserving transformations 37A05: Measure-preserving transformations
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 1 , January 2022 , pp. 86 - 140
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Footnotes

This paper is dedicated to Michael Boshernitzan, whose enthusiasm and passion remain an inspiration for mathematicians, both present and future.

References

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