No label defined (Q3314112)

closed additive subgroup of \({\mathbb{R}}\) generated by \(0<\eta<1\) and 1-\(\eta\). For irrational \(0<\theta<1\) consider the transformation \(T_{\theta,\eta}(x,y)=(x+\theta,y+\chi_{[0,\eta)}(x)-\eta)\quad in\quad {\mathbb{T}}\times {\mathbb{E}}_{\eta},\) where \(\chi_ A\) is the indicator function of A. Assume \(\eta\not\in {\mathbb{Z}}\theta\)(mod 1). It is shown that there exists a locally finite invariant measure