Dynamical behavior of alternate base expansions

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