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- Title
Rigidity of Entropy Spectra for One-Parameter Family of Polynomials.
- Authors
Nakagawa, Katsukuni
- 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.
- Subjects
ENTROPY; MARKOV spectrum; AUTOMORPHISM groups; POLYNOMIALS; OPEN spaces; EQUIVALENCE classes (Set theory); TOPOLOGICAL entropy
- Publication
Bulletin of the Brazilian Mathematical Society, 2022, Vol 53, Issue 2, p623
- ISSN
1678-7544
- Publication type
Article
- DOI
10.1007/s00574-021-00274-5