Shrinking random $\beta$-transformation

Abstract

For any $n\ge 3$, let $1<\beta <2$ be the largest positive real number satisfying the equation $${\beta }^n={\beta }^{n-2}+{\beta }^{n-3}+\cdots +\beta +1.$$ In this paper we define the shrinking random $\beta$-transformation $K$ and investigate natural invariant measures for $K$, and the induced tranformation of $K$ on a special subset of the domain. We prove that both transformations have a unique measure of maximal entropy. However, the measure induced from the intrinsically ergodic measure for $K$ is not the intrinsically ergodic measure for the induced system.

13 pages. To appear in Indagationes Mathematicae