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

Yan Huang, Lingmin Liao, Yun Sun

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 e…