Generic point equivalence and Pisot numbers - AKIYAMA, SHIGEKI; KANEKO, HAJIME; KIM, DONG HAN (Ergodic theory and dynamical systems, 2020)
Let $\unicode[STIX]{x1D6FD}>1$ be an integer or, generally, a Pisot number. Put $T(x)=\{\unicode[STIX]x1D6FDx\}$ on $[0,1]$ and let $S:[0,1]→ [0,1]$ be a piecewise linear transformation whose slopes have the form $± \unicode[STIX]{x1D6FD}^{m}$ with positive integers $m$. We give a sufficient condition for $T$ and $S$ to have the same generic points. We also give an uncountable family of maps which share the same set of generic points.