We study C 2 weakly order preserving circle maps with a flat interval. The main result of the paper is about a sharp transition from degenerate geometry to bounded geometry depending on the degree of the singularities at the boundary of the flat interval. We prove that the non-wandering set has zero Hausdorff dimension in the case of degenerate geometry and it has Hausdorff dimension strictly greater than zero in the case of bounded geometry. Our results about circle maps allow to establish a sharp phase transition in the dynamics of Cherry flows.
Newhouse laminations occur in unfoldings of rank-one homoclinic tangencies. Namely, in these unfoldings, there exist codimension 2 laminations of maps with infinitely many sinks which move simultaneously along the leaves. As consequence, in the space of real polynomial maps, there are examples of:-Hénon maps, in any dimension, with infinitely many sinks, -quadratic Hénon-like maps with infinitely many sinks and a period doubling attractor,-quadratic Hénon-like maps with infinitely many sinks and a strange attractor, -non trivial analytic families of polynomial maps with infinitely many sinks.
In this paper we study quasi-symmetric conjugations of C 2 weakly order-preserving circle maps with a flat interval. Under the assumption that the maps have the same rotation number of bounded type and that bounded geometry holds we construct a quasi-symmetric conjugation between their non-wandering sets. Further, this conjugation is extended to a quasi-symmetric circle homeomorphism. Our proof techniques hinge on real-dynamic methods allowing us to construct the conjugation under general and natural assumptions.
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