Shrinking random β-transformation

For any n≥3, let 1<β<2 be the largest positive real number satisfying the equation βn=βn−2+βn−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.