Matching for generalised $\beta$-transformations

TL;DR

This paper studies the property of matching in generalized beta-transformations, showing that for certain Pisot numbers, matching occurs densely and the size of the non-matching set is quantified, supported by numerical evidence.

Contribution

It demonstrates that for specific Pisot numbers, matching occurs on an open dense set of parameters and calculates the Hausdorff dimension of the non-matching set.

Findings

Matching occurs on an open dense set for certain Pisot numbers.

The Hausdorff dimension of the non-matching set is computed.

Numerical evidence suggests matching is widespread across parameters.

Abstract

We investigate matching for the family $T_\alpha(x) = \beta x + \alpha \pmod 1$, $\alpha \in [0,1]$, 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.