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On the number of ergodic measures for minimal shifts with eventually constant complexity growth

Published online by Cambridge University Press:  11 April 2016

MICHAEL DAMRON  and
JON FICKENSCHER
MICHAEL DAMRON
Affiliation:
School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA 30332, USA email [email protected]
JON FICKENSCHER
Affiliation:
Mathematics Department, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544, USA email [email protected]
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Abstract

In 1985, Boshernitzan showed that a minimal (sub)shift satisfying a linear block growth condition must have a bounded number of ergodic probability measures. Recently, this bound was shown to be sharp through examples constructed by Cyr and Kra. In this paper, we show that under the stronger assumption of eventually constant growth, an improved bound exists. To this end, we introduce special Rauzy graphs. Variants of the well-known Rauzy graphs from symbolic dynamics, these graphs provide an explicit description of how a Rauzy graph for words of length $n$ relates to the one for words of length $n+1$ for each $n=1,2,3,\ldots \,$.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 7 , October 2017 , pp. 2099 - 2130
Copyright
© Cambridge University Press, 2016 

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