Shrinking random β-transformation

Abstract

For any n ≥ 3, let 1 < β < 2 be the largest positive real number satisfying the equation β n = β n−2 + β n−3 + · · · + β + 1. In this paper we define the shrinking random β-transformation K and investigate natural invariant measures for K, and the induced transformation 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.