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

Abstract

We study the negative beta transformations $T_{-β}:=-βx +\lfloorβx\rfloor+1$ for $x\in(0,1]$ and $β>1$. We present a complete characterization of pairs of dstinct non-integers with the same $T_{-β}$-invariant measure: for two non-integers $β_1 ,β_2 >1$, the invariant measures of negative $β$-transformation coincide if and only if $β_1$ is the root of equation $x^2-qx-p=0$, where $p,q\in\mathbb{N}$ with $p\leq q$, and $β_2 = β_1 + 1$. Furthermore, we show that $T_{-β}$ has matching property for all $β$ being generalized multinacci numbers. We also prove that the set of simple $-β$ numbers, whose $-β$-shifts are subshifts of finite type, is dense in the parameter interval $(1,\infty)$.