The critical exponent functions
The critical exponent of a finite or infinite word $w$ over a given alphabet is the supremum of the reals $\alpha $ for which $w$ contains an $\alpha $-power. We study the maps associating to every real in the unit interval the inverse of the critical exponent of its base-$n$ expansion. We strengthe...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Académie des sciences
2022-04-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.286/ |
| _version_ | 1797651490629746688 |
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| author | Corona, Dario Della Corte, Alessandro |
| author_facet | Corona, Dario Della Corte, Alessandro |
| author_sort | Corona, Dario |
| collection | DOAJ |
| description | The critical exponent of a finite or infinite word $w$ over a given alphabet is the supremum of the reals $\alpha $ for which $w$ contains an $\alpha $-power. We study the maps associating to every real in the unit interval the inverse of the critical exponent of its base-$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. |
| first_indexed | 2024-03-11T16:16:33Z |
| format | Article |
| id | doaj.art-b3231d5233fa40e38be14064716f0861 |
| institution | Directory Open Access Journal |
| issn | 1778-3569 |
| language | English |
| last_indexed | 2024-03-11T16:16:33Z |
| publishDate | 2022-04-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj.art-b3231d5233fa40e38be14064716f08612023-10-24T14:19:45ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692022-04-01360G431533210.5802/crmath.28610.5802/crmath.286The critical exponent functionsCorona, Dario0https://orcid.org/0000-0002-7575-710XDella Corte, Alessandro1https://orcid.org/0000-0002-1782-0270University of Camerino, School of Science and Technology Camerino (MC), ItalyUniversity of Camerino,School of Science and Technology, Camerino (MC), ItalyThe critical exponent of a finite or infinite word $w$ over a given alphabet is the supremum of the reals $\alpha $ for which $w$ contains an $\alpha $-power. We study the maps associating to every real in the unit interval the inverse of the critical exponent of its base-$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.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.286/ |
| spellingShingle | Corona, Dario Della Corte, Alessandro The critical exponent functions Comptes Rendus. Mathématique |
| title | The critical exponent functions |
| title_full | The critical exponent functions |
| title_fullStr | The critical exponent functions |
| title_full_unstemmed | The critical exponent functions |
| title_short | The critical exponent functions |
| title_sort | critical exponent functions |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.286/ |
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