Dynamical properties of greedy and lazy alternate base expansions

In this presentation I study the greedy and lazy transformations in alternate bases β. These transformations can generate the digits of the corresponding greedy and lazy expansions. I first illustrate both transformations. Then I state the theorem on the existence
of a unique absolutely continuous (with respect to an extended Lebesgue measure, called
the p-Lebesgue measure) T_β-invariant measure. Next, I discuss on the dynamical properties of L_β by showing that the lazy dynamical system is isomorphic to the greedy one. I also prove an isomorphism with a suitable extension of the β-shift. Finally, I compare the β-expansions and the (βp−1 · · · β0)-representations over general digit sets. This is a joint work with Émilie Charlier and Karma Dajani.