On the invariant measure of the natural extension of the continued fraction transformation (Q1320518)

Let \(T\) be the classic transformation associated with the continued fraction expansions, that is, \(T(x) = \textstyle{1\over x} - [\textstyle{1\over x}]\), \(x\in (0,1]\) and \(T(0) = 0\). The author considers the natural extension \(S\) of \(T\) given by \[ S(x,y) = \biggl(Tx,{1\over a(x) + y}\biggr), \quad x,y\in (0,1), \] where \(a(x) = [\textstyle{1\over x}]\), and develops a new method to approach the invariant measure of \(S\).