Metric properties of non-renormalizable S-unimodal maps: II. Quasisymmetric conjugacy classes

Abstract

It is shown that for certain classes of S-unimodal maps with aperiodic kneading sequences, the topological conjugacies are also quasisymmetric. This includes some infinitely renormalizable polynomials of unbounded type.

Equilibrium measures for maps with inducing schemes

“…It is also shown there that the base W has full Lebesgue measure in A. Similar arguments work for any transverse family of unimodal maps using the fact that any non-renormalizable map of a full unimodal family is quasi-symmetrically conjugated to a map in the quadratic family (see [JŚ95]). Condition (H5) follows from Koebe's Distortion Lemma (see for example, [dMvS93]).…”

“…In particular, the Hausdorff dimension can thus be made arbitrarily small by choosing the number N 0 to be sufficiently large. In the general case, by [JŚ95], f a is Hölder conjugated to a quadratic map, so the Hausdorff dimension of ν can also be made arbitrarily small provided N 0 is sufficiently large.…”

Metric attractors for smooth unimodal maps

“…By Theorem 2, we may take the central domain of F to be as small as we like, proportionally to U , by taking U small enough. Adapting Proposition 5 from [7] to F , we see that F induces an expanding Markov map on some perhaps smaller regularly returning open interval U containing ζ, in other words f induces expansion on U .…”

Topological conditions for the existence of absorbing Cantor sets

“…If = 2, absorbing Cantor sets do not exist. Proofs were given by Lyubich [L1] and by Jakobson andŚwiatek [JS1], [JS2], [JS3]. However, as was proved in [BKNS], there are maps (with a degenerate critical point) that have an absorbing Cantor set.…”