Number theoretical transformations with finite range structure and their ergodic properties (Q1092104)

A fibered system (B,T) is called a number-theoretical transformation with finite range structure if the set of images \(T^ nB^{(n)}\) where \(B^{(n)}\) is a cylinder of rank n forms a finite partition of B. This notion covers various complex continued fractions, Jacobi-Perron algorithm and other multidimensional maps. The paper gives sufficient conditions for the existence of \(\sigma\)-finite invariant measures and ergodicity. Furthermore the maps \[ T_ 1(x,y)=(-[-1/x]-1/x,\quad -[- y/x]-y/x)\quad and\quad T_ 2(x,y)=(-[-1/x]-1/x,\quad y/x-[y/x]) \] are given special attention. Their invariant measures are given explicitly.