Geodesic
Measures of maximal entropy for random $\beta$-expansions
Journal of the European Mathematical Society, Tome 7 (2005) no. 1, pp. 51-68 Cet article a éte moissonné depuis la source EMS Press

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Let β>1 be a non-integer. We consider β-expansions of the form ∑i=1∞​βidi​​, where the digits (di​)i≥1​ are generated by means of a Borel map Kβ​ defined on {0,1}N×[0,⌊β⌋/(β−1)]. We show that Kβ​ has a unique mixing measure νβ​ of maximal entropy with marginal measure an infinite convolution of Bernoulli measures. Furthermore, under the measure νβ​ the digits (di​)i≥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 β-expansions.
DOI : 10.4171/jems/21
Classification : 28-XX, 00-XX
Keywords: greedy expansions, lazy expansions, Markov chains, measures of maximal entropy 
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     title = {Measures of maximal entropy for random $\beta$-expansions},
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     pages = {51--68},
     year = {2005},
     volume = {7},
     number = {1},
     doi = {10.4171/jems/21},
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Karma Dajani; Martijn de Vries. Measures of maximal entropy for random $\beta$-expansions. Journal of the European Mathematical Society, Tome 7 (2005) no. 1, pp. 51-68. doi: 10.4171/jems/21

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