Ergodic behavior of transformations associated with alternate base expansions

We consider a p-tuple of real numbers greater than 1, 𝛃=(𝛽_1,…,𝛽_p), called an alternate base, to represent real numbers. Since these representations generalize the 𝛽-representation introduced by Rényi in 1958, a lot of questions arise. In this talk, we will study the transformation generating the alternate base expansions (greedy representations). First, we will compare the 𝛃-expansion and the (𝛽_1*…*𝛽_p)-expansion over a particular digit set and study the cases when the equality holds. Next, we will talk about the existence of a measure equivalent to Lebesgue, invariant for the transformation corresponding to the alternate base and also about the ergodicity of this transformation.
This is a joint work with Émilie Charlier and Karma Dajani.