Negative $\beta$-transformations: invariant measures, subshifts of finite type and matching property

Abstract

We study the negative beta transformations $T_{-\beta }:=-\beta x+\lfloor \beta x\rfloor +1$ for $x\in (0,1]$ and $\beta >1$. We present a complete characterization of pairs of dstinct non-integers with the same $T_{-\beta }$-invariant measure: for two non-integers ${\beta }_1,{\beta }_2>1$, the invariant measures of negative $\beta$-transformation coincide if and only if ${\beta }_1$ is the root of equation $x^2-qx-p=0$, where $p,q\in \mathbb{N}$ with $p\le q$, and ${\beta }_2={\beta }_1+1$. Furthermore, we show that $T_{-\beta }$ has matching property for all $\beta$ being generalized multinacci numbers. We also prove that the set of simple $-\beta$ numbers, whose $-\beta$-shifts are subshifts of finite type, is dense in the parameter interval $(1,\infty )$.

19 pp