  For any $n\geq 3$, 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.
