Matching for generalised β-transformations

We investigate matching for the family Tα(x)=βx+α(mod1), α∈[0,1], for fixed β>1. Matching refers to the property that there is an n∈N such that Tαn(0)=Tαn(1). We show that for various Pisot numbers β, matching occurs on an open dense set of α∈[0,1] and we compute the Hausdorff dimension of its complement. Numerical evidence shows more cases where matching is prevalent.