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 β = (β0, . . . , βp−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 Tβ and Lβ respectively, can be iterated in order to
generate the digits of the greedy and lazy β-expansions of real numbers. The aim of this
paper is to describe the dynamical behaviors of Tβ and Lβ. We first prove the existence
of a unique absolutely continuous (with respect to an extended Lebesgue measure, called
the p-Lebesgue measure) Tβ-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 1/p log(βp−1 · · · β0). We then express the density of this p
measure and compute the frequencies of letters in the greedy β-expansions. We obtain the dynamical properties of Lβ 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 β-expansions can be seen as (βp−1 · · · β0 )-representations over general digit sets and we compare both frameworks.