Dynamical behavior of alternate base expansions

Abstract

We generalize the greedy and lazy $\beta$-transformations for a real base $\beta$ to the setting of alternate bases $\mathbf{\beta }=({\beta }_0,\dots ,{\beta }_{p-1})$, which were recently introduced by the first and second authors as a particular case of Cantor bases. As in the real base case, these new transformations, denoted $T_\boldsymbolβ$ and $L_\boldsymbolβ$ respectively, can be iterated in order to generate the digits of the greedy and lazy $\mathbf{\beta }$-expansions of real numbers. The aim of this paper is to describe the dynamical behaviors of $T_\boldsymbolβ$ and $L_\boldsymbolβ$. We first prove the existence of a unique absolutely continuous (with respect to an extended Lebesgue measure, called the $p$-Lebesgue measure) $T_\boldsymbolβ$-invariant measure. We then show that this unique measure is in fact equivalent to the $p$-Lebesgue measure and that the corresponding dynamical system is ergodic and has entropy $\frac{1}{p}\log ({\beta }_{p-1}\cdots {\beta }_0)$. We then express the density of this measure and compute the frequencies of letters in the greedy $\mathbf{\beta }$-expansions. We obtain the dynamical properties of $L_\boldsymbolβ$ by showing that the lazy dynamical system is isomorphic to the greedy one. We also provide an isomorphism with a suitable extension of the $\beta$-shift. Finally, we show that the $\mathbf{\beta }$-expansions can be seen as $({\beta }_{p-1}\cdots {\beta }_0)$-representations over general digit sets and we compare both frameworks.

28 pages, 15 figures