Dynamical behavior of alternate base expansions

Abstract

We generalize the greedy and lazy $\beta $ -transformations for a real base $\beta $ to the setting of alternate bases ${\boldsymbol {\beta }}=(\beta _0,\ldots ,\beta _{p-1})$ , 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 $T_{{\boldsymbol {\beta }}}$ and $L_{{\boldsymbol {\beta }}}$ respectively, can be iterated in order to generate the digits of the greedy and lazy ${\boldsymbol {\beta }}$ -expansions of real numbers. The aim of this paper is to describe the measure-theoretical dynamical behaviors of $T_{{\boldsymbol {\beta }}}$ and $L_{{\boldsymbol {\beta }}}$ . We first prove the existence of a unique absolutely continuous (with respect to an extended Lebesgue measure, called the p-Lebesgue measure) $T_{{\boldsymbol {\beta }}}$ -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 $({1}/{p})\log (\beta _{p-1}\cdots \beta _0)$ . We give an explicit expression of the density function of this invariant measure and compute the frequencies of letters in the greedy ${\boldsymbol {\beta }}$ -expansions. The dynamical properties of $L_{{\boldsymbol {\beta }}}$ 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 $\beta $ -shift. Finally, we show that the ${\boldsymbol {\beta }}$ -expansions can be seen as $(\beta _{p-1}\cdots \beta _0)$ -representations over general digit sets and we compare both frameworks.

Combinatorial properties of lazy expansions in Cantor real bases

“…In [5,Section 5], in the alternate base framework, both greedy and lazy expansions were compared. The following result generalizes this comparison to the Cantor base expansions.…”

“…In particular, generalizations of several combinatorial results of real base expansions were obtained, such as Parry's criterion for greedy expansions and, while considering periodic Cantor real bases, called alternate bases, Bertrand-Mathis' characterization of sofic shifts. Next, in [5], in the particular case of alternate bases, the lazy expansions were defined and both greedy and lazy expansions were studied in terms of dynamics. These results generalize the well-known ones from the theory of real base expansions (see [6,7,10,11]).…”

Spectrum, algebraicity and normalization in alternate bases

“…Representations of real numbers involving more than one base simultaneously and independently aroused the interest of other mathematicians [4,14,16,23]. But so far, most of the research was concentrated on the combinatorial properties of these representations as in [6] and not on their dynamical or algebraic properties which are studied respectively in [7] and in this paper.…”

“…Alternate bases are particular cases of Cantor real bases, which were introduced by the first two authors in [6] and then studied in [7,8]. A Cantor real base is a sequence β = (β n ) n∈N of real numbers greater than 1 such that +∞ n=0 β n = +∞.…”

“…In [8], the lazy combinatorial properties of Cantor real bases were investigated. Moreover, in [7], the classical dynamical results of greedy and lazy β-expansions were generalized while focusing on periodic Cantor real bases β = (β 0 , . .…”