The critical exponent functions

Dario Corona; Alessandro Della Corte

doi:10.5802/crmath.286

Combinatorics, Dynamical systems

The critical exponent functions

Dario Corona ¹ ; Alessandro Della Corte ²

¹ University of Camerino, School of Science and Technology Camerino (MC), Italy
² University of Camerino,School of Science and Technology, Camerino (MC), Italy

Comptes Rendus. Mathématique, Volume 360 (2022), pp. 315-332.

Abstract

The critical exponent of a finite or infinite word $w$ over a given alphabet is the supremum of the reals $α$ for which $w$ contains an $α$ -power. We study the maps associating to every real in the unit interval the inverse of the critical exponent of its base- $n$ expansion. We strengthen a combinatorial result by J.D. Currie and N. Rampersad to show that these maps are left- or right-Darboux at every point, and use dynamical methods to show that they have infinitely many nontrivial fixed points and infinite topological entropy. Moreover, we show that our model-case map is topologically mixing.

Received: 2021-09-04
Revised: 2021-10-13
Accepted: 2021-10-19
Published online: 2022-04-26

DOI: 10.5802/crmath.286

Classification: 37B40, 37B20, 68R15, 26A21, 26A18

Author's affiliations:

Dario Corona ¹; Alessandro Della Corte ²

¹ University of Camerino, School of Science and Technology Camerino (MC), Italy
² University of Camerino,School of Science and Technology, Camerino (MC), Italy

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Dario Corona and Alessandro Della Corte},
     title = {The critical exponent functions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {315--332},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {360},
     year = {2022},
     doi = {10.5802/crmath.286},
     language = {en},
}

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Dario Corona; Alessandro Della Corte. The critical exponent functions. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 315-332. doi : 10.5802/crmath.286. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.286/

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