Invariant densities for random <em>β</em>-expansions

  • Karma Dajani

    Universiteit Utrecht, Netherlands
  • Martijn de Vries

    Vrije Universiteit, Amsterdam, Netherlands


Let β>1\beta >1 be a non-integer. We consider expansions of the form i=1diβi\sum_{i=1}^{\infty} \frac{d_i}{\beta^i}, where the digits (di)i1(d_i)_{i \geq 1} are generated by means of a Borel map KβK_{\beta} defined on {0,1}N×[0,β/(β1)]\{0,1\}^{\N}\times \left[ 0, \lfloor \beta \rfloor /(\beta -1)\right]. We show existence and uniqueness of an absolutely continuous KβK_{\beta}-invariant probability measure w.r.t. mpλm_p \otimes \lambda, where mpm_p is the Bernoulli measure on {0,1}N\{0,1\}^{\N} with parameter pp (0<p<1)( 0 < p < 1) and λ\lambda is the normalized Lebesgue measure on [0,β/(β1)][0 ,\lfloor \beta \rfloor /(\beta -1)]. Furthermore, this measure is of the form mpμβ,pm_p \otimes \mu_{\beta,p}, where μβ,p\mu_{\beta,p} is equivalent with λ\lambda. We establish the fact that the measure of maximal entropy and mpλm_p \otimes \lambda are mutually singular. In case 11 has a finite greedy expansion with positive coefficients, the measure mpμβ,pm_p \otimes \mu_{\beta,p} is Markov. In the last section we answer a question concerning the number of universal expansions, a notion introduced in [EK].

Cite this article

Karma Dajani, Martijn de Vries, Invariant densities for random <em>β</em>-expansions. J. Eur. Math. Soc. 9 (2007), no. 1, pp. 157–176

DOI 10.4171/JEMS/76