Generic point equivalence and Pisot numbers

Let $\beta >1$ be an integer or generally a Pisot number. Put $T(x) = { \beta x }$on$[0,1]$andlet$S: [0,1]\to [0,1]$ be a piecewise linear transformation whose slopes have the form $\pm {\beta }^m$ with positive integers $m$. We give sufficient conditions that $T$ and $S$ have the same generic points.