Matching for linear mod one transformations

For a linear mod one transformation (also called intermediate beta-transformation) $T_{\beta,\alpha}(x) = \beta x + \alpha - \lfloor \beta x + \alpha\rfloor$, matching holds if $T_{\beta,\alpha}^n(0) = T_{\beta,\alpha}^n(1^-)$ for some $n$. In this case, the absolutely continuous invariant measure has piecewise constant density. Bruin, Carminati and Kalle (2017) proved that matching occurs for almost all $\alpha$ when the base $\beta$ is a quadratic Pisot number or the Tribonacci number, and they conjecture that this holds for all Pisot numbers. Sun, Li and Ding (2023) have established relations with intermediate $\beta$-shifts of finite type and proved results on the fiber density. We discuss for which bases matching can occur and show that 0 is an accumulation point of matching parameters when $\beta$ is a Pisot number or a simple Parry number. We also discuss matching properties for $\alpha$-continued fractions.