A complex continued fraction transformation and its ergodic properties
Denote by X the set of all complex numbers z of the form \(z=x\alpha +y{\bar \alpha}\), where \(x,y\in [-1/2,1/2]\) and \(\alpha =1+i\). In this paper the mapping T is defined in the following way: \(Tz=1/z-[1/z]\), where \([z_ 1]_ 1=[x+1/2]\alpha +[y+1/2]{\bar \alpha}\) for \(z=x\alpha +y{\bar \alpha}\). This mapping induces a continued fraction expansion for \(z\in X,\) \(z=\frac{1|}{| \alpha_ 1}+\frac{1|}{| \alpha_ 2}+...\quad,\)where \(\alpha_ j\) \((j=1,2...)\) has the form \(n\alpha +m{\bar \alpha}\) with n,m integers. Simultaneously a dual transformation S and a natural extension R of T is introduced here. Exact forms of absolutely continuous invariant measures for T and S are determined. Also the ergodicity of T and S with respect to a certain introduced invariant measure is studied.
scientific article; zbMATH DE number 3929104