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Shrinking random $β$-transformation

For any $n\geq 3$, let $1<β<2$ be the largest positive real number satisfying the equation $$β^n=β^{n-2}+β^{n-3}+\cdots+β+1.$$ In this paper we define the shrinking random $β$-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.