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Invariant density functions of random $\unicode[STIX]{x1D6FD}$-transformations

Published online by Cambridge University Press:  07 September 2017

SHINTARO SUZUKI
SHINTARO SUZUKI*
Affiliation:
Department of Mathematics, Graduate School of Science, Osaka University, 1-1 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan email si-suzuki@cr.math.sci.osaka-u.ac.jp
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Abstract

We consider the random $\unicode[STIX]{x1D6FD}$-transformation $K_{\unicode[STIX]{x1D6FD}}$ introduced by Dajani and Kraaikamp [Random $\unicode[STIX]{x1D6FD}$-expansions. Ergod. Th. & Dynam. Sys.23 (2003), 461–479], which is defined on $\{0,1\}^{\mathbb{N}}\times [0,[\unicode[STIX]{x1D6FD}]/(\unicode[STIX]{x1D6FD}-1)]$. We give an explicit formula for the density function of a unique $K_{\unicode[STIX]{x1D6FD}}$-invariant probability measure absolutely continuous with respect to the product measure $m_{p}\otimes \unicode[STIX]{x1D706}_{\unicode[STIX]{x1D6FD}}$, where $m_{p}$ is the $(1-p,p)$-Bernoulli measure on $\{0,1\}^{\mathbb{N}}$ and $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D6FD}}$ is the normalized Lebesgue measure on $[0,[\unicode[STIX]{x1D6FD}]/(\unicode[STIX]{x1D6FD}-1)]$. We apply the explicit formula for the density function to evaluate its upper and lower bounds and to investigate its continuity as a function of the two parameters $p$ and $\unicode[STIX]{x1D6FD}$.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 4 , April 2019 , pp. 1099 - 1120
Copyright
© Cambridge University Press, 2017 

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