Let \(a_n\) (\(n\geq 1\)) be the partial quotients of the continued fraction expansion of an irrational number \(\omega\) in the unit interval \(I\). If we use the transformation \(\tau(\omega)=1/\omega\mod 1\), then \(\omega=[0;a_1,\dots,a_n+\tau^n(\omega)]\) for any \(n\geq 1\). Let \(\gamma_a\) (\(a\in I\)) be the probability measure on the Borel sets of \(I\) defined by \(\gamma_a([0,x])=(a+1)x/(ax+1)\) (\(x\in I\)). The authors obtained in [Metrical theory of continued fractions, Mathematics and its Applications 547, Dordrecht: Kluwer Academic Publishers (2002; Zbl 1069.11032)] that for any \(a\in I\) and any positive integer~\(n\), \[ \gamma_a(\tau^n