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• W4206576006 abstract "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." @default.
• W4206576006 created "2022-01-25" @default.
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• W4206576006 date "2021-12-22" @default.
• W4206576006 modified "2023-10-18" @default.
• W4206576006 title "Dynamical behavior of alternate base expansions" @default.
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• W4206576006 doi "https://doi.org/10.1017/etds.2021.161" @default.
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