# Invariant density functions of random β-transformations

Research output: Contribution to journalArticle

### Abstract

We consider the random -transformation introduced by Dajani and Kraaikamp [Random -expansions. Ergod. Th. & Dynam. Sys. 23 (2003), 461-479], which is defined on . We give an explicit formula for the density function of a unique -invariant probability measure absolutely continuous with respect to the product measure , where is the -Bernoulli measure on and is the normalized Lebesgue measure on . 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 and .

Original language English 1099-1120 22 Ergodic Theory and Dynamical Systems 39 4 https://doi.org/10.1017/etds.2017.64 Published - 2019 Apr 1 Yes

### Fingerprint

Density Function
Probability density function
Explicit Formula
Product Measure
Invariant
Lebesgue Measure
Absolutely Continuous
Bernoulli
Invariant Measure
Probability Measure
Two Parameters
Upper and Lower Bounds
Evaluate

### ASJC Scopus subject areas

• Mathematics(all)
• Applied Mathematics

### Cite this

In: Ergodic Theory and Dynamical Systems, Vol. 39, No. 4, 01.04.2019, p. 1099-1120.

Research output: Contribution to journalArticle

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abstract = "We consider the random -transformation introduced by Dajani and Kraaikamp [Random -expansions. Ergod. Th. & Dynam. Sys. 23 (2003), 461-479], which is defined on . We give an explicit formula for the density function of a unique -invariant probability measure absolutely continuous with respect to the product measure , where is the -Bernoulli measure on and is the normalized Lebesgue measure on . 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 and .",
author = "Shintaro Suzuki",
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AB - We consider the random -transformation introduced by Dajani and Kraaikamp [Random -expansions. Ergod. Th. & Dynam. Sys. 23 (2003), 461-479], which is defined on . We give an explicit formula for the density function of a unique -invariant probability measure absolutely continuous with respect to the product measure , where is the -Bernoulli measure on and is the normalized Lebesgue measure on . 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 and .

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