Negative $\beta$-transformations: invariant measures, matching properties and associated subshifts

Abstract

For a real number $\beta >1$, we study the negative $\beta$-transformations, defined by $T_{-\beta }(x):=-\beta x+\lfloor \beta x\rfloor +1$ for $x\in (0,1]$. It is known that for each $\beta >1$, there exists a unique $T_{-\beta }$-invariant probability measure which is absolutely continuous with respect to the Lebesgue measure, denoted by ${\nu }_{-\beta }$. For two distinct non-integer real numbers ${\beta }_2>{\beta }_1>1$, we prove that ${\nu }_{-{\beta }_1}={\nu }_{-{\beta }_2}$ if and only if ${\beta }_1$ is a root of the quadratic equation $x^2-qx-p=0$ (where $p,q\in \mathbb{N}$ and $p\le q$) and ${\beta }_2={\beta }_1+1$. Furthermore, we show that $T_{-\beta }$ satisfies the matching property for all generalized multinacci numbers $\beta$. Finally, we prove that the simple $(-\beta )$-numbers, i.e., the numbers whose corresponding $(-\beta )$-shifts are subshifts of finite type, are dense in the parameter space $(1,\infty )$.

arXiv comment22 pp; Author list updated in v2