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Rigidity of Entropy Spectra for One-Parameter Family of Polynomials

Rigidity of Entropy Spectra for One-Parameter Family of Polynomials In this paper, we consider entropy spectra of Markov measures on topological Markov shifts. For a Markov measure on a shift, using the description of its entropy spectrum via matrix theory, we naturally obtain a one-parameter family of polynomials. We consider the set of all Markov measures on the shift whose entropy spectra have all information about the families of polynomials, unlike Barreira and Saraiva who considered the set of those whose entropy spectra have all information about the equivalence classes of the relation induced by the action of the automorphism group of the shift. We give a sufficient condition on a shift for our set to contain a full-measure open set of the space of all Markov measures on the shift. This rigidity result is in contrast to the non-rigidity result of Barreira and Saraiva that, for a certain topological Markov shift, the complement of their set contains a full-measure open set of the space of all Markov measures on the shift. The shift of Barreira and Saraiva satisfies our condition, and hence, the shift turns out to be rigid in our sense. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png "Bulletin of the Brazilian Mathematical Society, New Series" Springer Journals

Rigidity of Entropy Spectra for One-Parameter Family of Polynomials

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References (11)

Publisher
Springer Journals
Copyright
Copyright © Sociedade Brasileira de Matemática 2021
ISSN
1678-7544
eISSN
1678-7714
DOI
10.1007/s00574-021-00274-5
Publisher site
See Article on Publisher Site

Abstract

In this paper, we consider entropy spectra of Markov measures on topological Markov shifts. For a Markov measure on a shift, using the description of its entropy spectrum via matrix theory, we naturally obtain a one-parameter family of polynomials. We consider the set of all Markov measures on the shift whose entropy spectra have all information about the families of polynomials, unlike Barreira and Saraiva who considered the set of those whose entropy spectra have all information about the equivalence classes of the relation induced by the action of the automorphism group of the shift. We give a sufficient condition on a shift for our set to contain a full-measure open set of the space of all Markov measures on the shift. This rigidity result is in contrast to the non-rigidity result of Barreira and Saraiva that, for a certain topological Markov shift, the complement of their set contains a full-measure open set of the space of all Markov measures on the shift. The shift of Barreira and Saraiva satisfies our condition, and hence, the shift turns out to be rigid in our sense.

Journal

"Bulletin of the Brazilian Mathematical Society, New Series" – Springer Journals

Published: Jun 1, 2022

Keywords: Entropy spectra; Rigidities of entropy spectra; Markov measures; Primary 37A35; Secondary 37B10; 37B40

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