Matching for generalised $\beta$-transformations

Abstract

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

16 pages, 11 figures