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

For a real number $β>1$, we study the negative $β$-transformations, defined by $T_{-β}(x) := -βx + \lfloorβx\rfloor + 1$ for $x \in (0,1]$. It is known that for each $β>1$, there exists a unique $T_{-β}$-invariant probability measure which is absolutely continuous with respect to the Lebesgue measur