The critical exponent functions

Abstract

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.