Given a real number \(\beta>1\), the \(\beta\)-transformation is the selfmap \(T_{\beta}\) of \([0,1)\) defined by \(T_{\beta}(x)=\beta x-\lfloor\beta x\rfloor\), while the negative \(\beta\)-transformation is the selfmap \(T_{-\beta}\) of \((0,1]\) defined by \(T_{-\beta}(x)=-\beta x+\lfloor\beta x\rfloor+1\). The two maps are related by \(T_{-\beta}=1-T_{\beta}\) on \((0,1)\); they are both expanding, piecewise monotonic, and have a unique absolutely continuous invariant measure.
While in the case of the \(\beta\)-transformation the invariant density is never \(0\), in the case of \(T_{-\beta}\) the invariant density may be \(0\) on finitely many subintervals. Writing \(G(\beta)\) for the union of these subintervals – which are explicitly described – the main result of this paper states that \(T_{-\beta}\) is topologically exact on \((0,1]\setminus G(\beta)\). This is the topological analogue of Rohlin’s exactness property: for every nonempty open set \(U\subseteq (0,1]\setminus G(\beta)\) there exists an \(n\) such that \(T_{-\beta}^n(U)=(0,1]\setminus G(\beta)\).
From the above result the authors derive various consequences, extending previous works by Faller, Góra, Ito-Sadahiro, and Masáková-Pelantová. In particular, they prove that \(T_{-\beta}\) is Rohlin exact, that its maximal-entropy measure is unique, and that every \(\beta\) for which the \(T_{-\beta}\)-orbit of \(1\) is eventually periodic is a Perron number.