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Invariant densities for random β -expansions

Karma DajaniMartijn de Vries — 2007

Journal of the European Mathematical Society

Let β > 1 be a non-integer. We consider expansions of the form i = 1 d i / β i , where the digits ( d i ) i 1 are generated by means of a Borel map K β defined on { 0 , 1 } × [ 0 , β ( β 1 ) ] . We show existence and uniqueness of a K β -invariant probability measure, absolutely continuous with respect to m p λ , where m p is the Bernoulli measure on { 0 , 1 } with parameter p ( 0 < p < 1 ) and λ is the normalized Lebesgue measure on [ 0 , β ( β 1 ) ] . Furthermore, this measure is of the form m p μ β , p , where μ β , p is equivalent to λ . We prove that the measure of maximal entropy and m p λ are mutually singular. In...

Measures of maximal entropy for random β -expansions

Karma DajaniMartijn de Vries — 2005

Journal of the European Mathematical Society

Let β > 1 be a non-integer. We consider β -expansions of the form i = 1 d i / β i , where the digits ( d i ) i 1 are generated by means of a Borel map K β defined on { 0 , 1 } × [ 0 , β / ( β 1 ) ] . We show that K β has a unique mixing measure ν β of maximal entropy with marginal measure an infinite convolution of Bernoulli measures. Furthermore, under the measure ν β the digits ( d i ) 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....

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