Dynamical behavior of alternate base expansions
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We generalize the greedy and lazy β-transformations for a real base β to the setting of alternate bases β=(β_0,…,β_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_β and L_β respectively, can be iterated in order to generate the digits of the greedy and lazy β-expansions of real numbers. The aim of this paper is to describe the dynamical behaviors of T_β and L_β. We first prove the existence of a unique absolutely continuous (with respect to an extended Lebesgue measure, called the p-Lebesgue measure) T_β-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/plog(β_p-1⋯β_0). We then express the density of this measure and compute the frequencies of letters in the greedy β-expansions. We obtain the dynamical properties of L_β by showing that the lazy dynamical system is isomorphic to the greedy one. We also provide an isomorphism with a suitable extension of the β-shift. Finally, we show that the β-expansions can be seen as (β_p-1⋯β_0)-representations over general digit sets and we compare both frameworks.
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