Combinatorics, Dynamical systems
The critical exponent functions
Comptes Rendus. Mathématique, Volume 360 (2022) no. G4, pp. 315-332.

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:
Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.286
Classification: 37B40, 37B20, 68R15, 26A21, 26A18
Corona, Dario 1; Della Corte, Alessandro 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
@article{CRMATH_2022__360_G4_315_0,
     author = {Corona, Dario and Della Corte, Alessandro},
     title = {The critical exponent functions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {315--332},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {360},
     number = {G4},
     year = {2022},
     doi = {10.5802/crmath.286},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/crmath.286/}
}
TY  - JOUR
AU  - Corona, Dario
AU  - Della Corte, Alessandro
TI  - The critical exponent functions
JO  - Comptes Rendus. Mathématique
PY  - 2022
SP  - 315
EP  - 332
VL  - 360
IS  - G4
PB  - Académie des sciences, Paris
UR  - http://www.numdam.org/articles/10.5802/crmath.286/
DO  - 10.5802/crmath.286
LA  - en
ID  - CRMATH_2022__360_G4_315_0
ER  - 
%0 Journal Article
%A Corona, Dario
%A Della Corte, Alessandro
%T The critical exponent functions
%J Comptes Rendus. Mathématique
%D 2022
%P 315-332
%V 360
%N G4
%I Académie des sciences, Paris
%U http://www.numdam.org/articles/10.5802/crmath.286/
%R 10.5802/crmath.286
%G en
%F CRMATH_2022__360_G4_315_0
Corona, Dario; Della Corte, Alessandro. The critical exponent functions. Comptes Rendus. Mathématique, Volume 360 (2022) no. G4, pp. 315-332. doi : 10.5802/crmath.286. http://www.numdam.org/articles/10.5802/crmath.286/

[1] Adamczewski, Boris On the expansion of some exponential periods in an integer base, Math. Ann., Volume 346 (2010), pp. 107-116 | DOI | MR | Zbl

[2] Adamczewski, Boris; Bugeaud, Yann Dynamics for β-shifts and Diophantine approximation, Ergodic Theory Dyn. Syst., Volume 27 (2007) no. 6, pp. 1695-1711 | DOI | MR | Zbl

[3] Allouche, Jean-Paul; Shallit, Jeffrey The ubiquitous Prouhet–Thue–Morse sequence, Sequences and their applications. Proceedings of the international conference, SETA ’98, Singapore, December 14-17, 1998 (Ding, C. et al., eds.) (Springer Series in Discrete Mathematics and Theoretical Computer Science), Springer, 1999, pp. 1-16 | Zbl

[4] Baire, René Leçons sur les fonctions discontinues, Gauthier-Villars, 1905 (VIII u. 127 S.8°.) | Zbl

[5] Berstel, Jean; Lauve, Aaron; Reutenauer, Christophe; Saliola, Franco V. Combinatorics on Words: Christoffel Words and Repetitions in Words, CRM Monograph Series, 27, American Mathematical Society, 2008 | DOI | Zbl

[6] Bowen, Rufus Entropy for group endomorphisms and homogeneous spaces, Trans. Am. Math. Soc., Volume 153 (1971), pp. 401-414 | DOI | MR | Zbl

[7] Currie, James D.; Rampersad, Narad For Each α> 2 There Is an Infinite Binary Word with Critical Exponent α, Electron. J. Comb. (2008), N34 | MR | Zbl

[8] Currie, James D.; Rampersad, Narad A proof of Dejean’s conjecture, Math. Comput., Volume 80 (2011) no. 274, pp. 1063-1070 | DOI | MR | Zbl

[9] Dejean, Françoise Sur Un Théorème de Thue, J. Comb. Theory, Ser. A, Volume 13 (1972) no. 1, pp. 90-99 | DOI | MR | Zbl

[10] Dinaburg, Efim Isaakovich A correlation between topological entropy and metric entropy, Dokl. Akad. Nauk SSSR, Volume 190 (1970) no. 1, pp. 19-22 | MR | Zbl

[11] Falconer, Kenneth Fractal geometry: mathematical foundations and applications, John Wiley & Sons, 2003 | DOI | Zbl

[12] Filip, Ferdinánd; Šustek, Jan An elementary proof that almost all real numbers are normal, Acta Univ. Sapientiae, Math., Volume 2 (2010), pp. 99-110 | MR | Zbl

[13] Karhumäki, Juhani; Shallit, Jeffrey Polynomial versus exponential growth in repetition-free binary words, J. Comb. Theory, Ser. A, Volume 105 (2004) no. 2, pp. 335-347 | DOI | MR | Zbl

[14] Krieger, Dalia; Shallit, Jeffrey Every Real Number Greater than 1 Is a Critical Exponent, Theor. Comput. Sci., Volume 381 (2007) no. 1, pp. 177-182 | DOI | MR | Zbl

[15] Liouville, Joseph Sur des classes très-étendues de quantités dont la valeur n’est ni algébrique, ni même réductible à des irrationnelles algébriques., J. Math. Pures Appl. (1851), pp. 133-142

[16] Pawlak, Ryszard J. On the Entropy of Darboux Functions, Colloq. Math., Volume 116 (2009) no. 2, pp. 227-241 | DOI | MR | Zbl

[17] Queffelec, Martine Old and new results on normality, Dynamics and stochastics. Festschrift in honor of M. S. Keane (Denteneer, Dee et al., eds.) (Institute of Mathematical Statistics Lecture Notes - Monograph Series), Volume 48, Institute of Mathematical Statistics, 2006, pp. 225-236 | DOI | MR | Zbl

[18] Rao, Michaël Last Cases of Dejean’s Conjecture, Theor. Comput. Sci., Volume 412 (2011) no. 27, pp. 3010-3018 | DOI | MR | Zbl

[19] Rosen, Harvey Darboux quasicontinuous functions, Real Anal. Exch., Volume 23(1997-98) (1998) no. 2, pp. 631-639 | MR | Zbl

[20] Ruette, Sylvie Chaos on the Interval, University Lecture Series, 67, American Mathematical Society, 2017 | MR | Zbl

[21] Steele, Timothy H. Dynamics of Typical Baire-1 Functions on the Interval, J. Appl. Anal., Volume 23 (2017) no. 2, pp. 59-64 | MR | Zbl

[22] Steele, Timothy H. Dynamics of Baire-1 Functions on the Interval, Eur. J. Math., Volume 5 (2019) no. 1, pp. 138-149 | DOI | MR | Zbl

[23] Thue, Axel Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen, J. Dybwad, 1912 (67 S. Lex. OCLC: 458299532) | Zbl

[24] Vaslet, Elise Critical Exponents of Words over 3 Letters, Electron. J. Comb. (2011), P125 | DOI | MR | Zbl

Cited by Sources: