Dynamical behavior of alternate base expansions

We generalize the greedy and lazy $β$-transformations for a real base $β$ to the setting of alternate bases $\boldsymbolβ=(β_0,\ldots,β_{p-1})$, 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 $\boldsymbolβ$-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 $\frac{1}{p}\log(β_{p-1}\cdots β_0)$. We then express the density of this measure and compute the frequencies of letters in the greedy $\boldsymbolβ$-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 $β$-shift. Finally, we show that the $\boldsymbolβ$-expansions can be seen as $(β_{p-1}\cdots β_0)$-representations over general digit sets and we compare both frameworks.

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