Generic point equivalence and Pisot numbers

Abstract

Let beta > 1 be an integer or, generally, a Pisot number. Put T (x) = {beta x} on [0, 1] and let S : [0; 1] -> [0, 1] be a piecewise linear transformation whose slopes have the form +/- beta(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.