Invariant measures for parabolic IFS with overlaps and random continued fractions (Q2750939)

Let \(\nu_\alpha\) be the distribution of the random continued fraction \([1,Y_1,1,Y_2,1,Y_3,\dots]\) where the \(Y_i\) are i.i.d. and \(Y_i=0\) or \(\alpha\) with probabilities \(\frac 12\) each. Then \(\nu_\alpha\) is supported on a Cantor set of zero Lebesgue measure for \(\alpha>0.5\), hence singular, and supported on an interval for \(\alpha\leq 0.5\). \textit{R. Lyons} has proved recently [J. Theor. Probab. 13, No. 2, 535-545 (2000; Zbl 0971.60021)] that \(\nu_\alpha\) is moreover singular for all \(\alpha\in (\alpha_c,0.5]\) where \(\alpha_c \approx 0.2688\) is an explicitly given constant. The authors prove that this bound is sharp: \(\nu_\alpha\) is absolutely continuous for a.e. \(\alpha \in (\alpha_0,\alpha_c)\) for some \(\alpha_0\leq 0.215\). This follows in fact from a much more general theorem on invariant measures of parabolic iterated function systems (IFS) which the authors prove here, using methods and ideas they have developed earlier. Of central importance is the transversality condition, giving conditions on the dependence of the IFS on \(\alpha\). This condition has been introduced by the authors and their coworkers in different frameworks and formulations, in papers such as Ann. Math. (2) 142, No. 3, 611-625 (1995; Zbl 0837.28007) or Trans. Am. Math. Soc. 350, No. 10, 4065-4087 (1998; Zbl 0912.28005).