Measures of maximal entropy for random B-expansions

• Autores: Martijn de Vries, Karma Dajani
• Localización: Journal of the European Mathematical Society, ISSN 1435-9855, Vol. 7, Nº 1, 2005, págs. 51-68
• Idioma: inglés
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• Resumen

  • Let $\beta >1$ be a non-integer. We consider $\beta$-expansions of the form $\sum_{i=1}^{\infty} \frac{d_i}{\beta^i}$, where the digits $(d_i)_{i \geq 1}$ are generated by means of a Borel map $K_{\beta}$ defined on $\{0,1\}^{\N}\times \left[ 0, \lfloor \beta \rfloor /(\beta -1)\right]$. We show that $K_{\beta}$ has a unique mixing measure $\nu_{\beta}$ of maximal entropy with marginal measure an infinite convolution of Bernoulli measures. Furthermore, under the measure $\nu_{\beta}$ the digits $(d_i)_{i \geq 1}$ form a uniform Bernoulli process. In case 1 has a finite greedy expansion with positive coefficients, the measure of maximal entropy is Markov. We also discuss the uniqueness of $\beta$-expansions.