Invariant densities for random $β$-expansions
| dc.creator | Dajani, K. | |
| dc.creator | de Vries, M. | |
| dc.date | 2006-12-20 | |
| dc.date.accessioned | 2026-07-07T07:36:22Z | |
| dc.date.available | 2026-07-07T07:36:22Z | |
| dc.description | Let $β>1$ be a non-integer. We consider expansions of the form $\sum_{i=1}^{\infty} d_i β^{-i}$, where the digits $(d_i)_{i \geq 1}$ are generated by means of a Borel map $K_β$ defined on $\{0,1\}^{\N}\times [ 0, \lfloor β\rfloor /(β-1)]$. We show existence and uniqueness of an absolutely continuous $K_β$-invariant probability measure w.r.t. $m_p \otimes λ$, where $m_p$ is the Bernoulli measure on $\{0,1\}^{\N}$ with parameter $p$ $(0 < p < 1)$ and $λ$ is the normalized Lebesgue measure on $[0 ,\lfloor β\rfloor /(β-1)]$. Furthermore, this measure is of the form $m_p \otimes μ_{β,p}$, where $μ_{β,p}$ is equivalent with $λ$. We establish the fact that the measure of maximal entropy and $m_p \otimes λ$ are mutually singular. In case the number 1 has a finite greedy expansion with positive coefficients, the measure $m_p \otimes μ_{β,p}$ is Markov. In the last section we answer a question concerning the number of universal expansions, a notion introduced in [EK]. | |
| dc.identifier | https://arxiv.org/abs/math/0612602 | |
| dc.identifier | http://arxiv.org/abs/math/0612602 | |
| dc.identifier | J. Eur. Math. Soc. 7 (2007), 157--176 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/120399 | |
| dc.subject | Dynamical Systems | |
| dc.subject | Number Theory | |
| dc.subject | Primary: 28D05, Secondary: 11K16, 28D20, 37A35, 37A45 | |
| dc.title | Invariant densities for random $β$-expansions | |
| dc.type | text |