Ergodic behavior of transformations associated with alternate base expansions

Abstract: We consider a p-tuple of real numbers greater than 1, $\boldsymbol{\beta} = (\beta_1,\dots,\beta_p)$, 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 $\boldsymbol{\beta}$-expansion and the $(\beta_1*\cdots*\beta_p)$-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.

dynamical systemsnumber theory

Audience: researchers in the topic

One World Numeration seminar

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