The critical exponent functions

Corona, Dario; Della Corte, Alessandro

Comptes Rendus. Mathématique, Tome 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 $\alpha$ 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.

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

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