Some skew product transformations associated with continued fractions and their invariant measures (Q1084443)

The paper is a careful study of the ergodic properties of the maps \[ T_ 1(x,y)=(1/x-[1/x], y/x-[y/x])\quad and\quad T_ 2(x,y)=(1/x-[1/x], -y/x-[-y/x]). \] These transformations induce related expansions of irrational numbers y in the form \(y=\sum^{\infty}_{k=1}| \theta (k-1)| b(k),\) and \(y=\sum^{\infty}_{k=1}\theta (k-1) c(k)\) where \(\theta (n)=q_ n x-p_ n\) which are useful for estimating the discrepancy of the sequence \(\{\) nx\(\}\) [see \textit{V. T. Sós}, Top. Number Theory, Debrecen 1974, Colloq. Math. Soc. János Bolyai 13, 359- 367 (1976; Zbl 0354.10028)].