Matching for generalised $β$-transformations

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

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