Measure of solenoidal attractors of unimodal maps of the segment (Q1814629)

Let \(f\) be a \(C^ 3\) \(S\)-unimodal map from the unit interval to itself with a solenoidal (``Feigenbaum-type'') attractor \(A\) which corresponds to a sequence of periodic orbits with periods \(p_ 1,p_ 1p_ 2\), \(p_ 1p_ 2p_ 3,\dots\). It is shown that \(A\) has Lebesgue measure zero. If \(p_{k+1}/p_ k\) is bounded above, \(A\) has Hausdorff dimension \(<1\). In a recent paper, the authors succeeded in proving this theorem for arbitrary multimodal \(C^ \infty\) maps with nonflat critical points [Commun. Math. Phys. 127, No. 3, 573-583 (1990; Zbl 0721.58033)]. Similar results have been given in a preprint by Martens, de Melo, van Strien and Sullivan, and by \textit{J. Guckenheimer} and \textit{S. Johnson}, who gave a survey of related questions in their well-written introduction [Ann. Math., II. Ser. 132, No. 1, 71-130 (1990; Zbl 0708.58007)].