Dynamical behavior of alternate base expansions

We generalize the greedy and lazy -transformations for a real base to the setting of alternate bases, 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 and respectively, can be iterated in order to generate the digits of the greedy and lazy -expansions of real numbers. The aim of this paper is to describe the measure-theoretical dynamical behaviors of and. We first prove the existence of a unique absolutely continuous (with respect to an extended Lebesgue measure, called the p-Lebesgue measure) -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. We give an explicit expression of the density function of this invariant measure and compute the frequencies of letters in the greedy -expansions. The dynamical properties of are obtained by showing that the lazy dynamical system is isomorphic to the greedy one. We also provide an isomorphism with a suitable extension of the -shift. Finally, we show that the -expansions can be seen as -representations over general digit sets and we compare both frameworks.