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Global results are proved about the way in which Boyland's forcing partial order organizes a set of braid types: those of periodic orbits of Smale's horseshoe map for which the associated train track is a star. This is a special case of a conjecture introduced in [1], which claims that forcing organizes all horseshoe braid types into linearly ordered families which are, in turn, parametrized by homoclinic orbits to the fixed point of code 0.
This paper investigates the set of angles of the parameter rays which land on the real slice [-2, 1/4] of the Mandelbrot set. We prove that this set has zero length but Hausdorff dimension 1. We obtain the corresponding results for the tuned images of the real slice. Applications of these estimates in the study of critically non-recurrent real quadratics as well as biaccessible points of quadratic Julia sets are given.
In this work, under a mild assumption, we give the classification of the complete polynomial vector fields in two variables up to algebraic automorphisms of $\mathbb{C}^2$. The general problem is also reduced to the study of the combinatorics of certain resolutions of singularities. Whereas we deal with $\mathbb{C}$-complete vector fields, our results also apply to $\mathbb{R}$-complete ones thanks to a theorem of Forstneric [Fo].
A homeomorphism of a compact metric space is tight provided every non-degenerate compact connected (not necessarily invariant) subset carries positive entropy. It is shown that every $C^{1+\alpha}$ diffeomorphism of a closed surface factors to a tight homeomorphism of a generalized cactoid (roughly, a surface with nodes) by a semi-conjugacy whose fibers carry zero entropy.
David maps are generalizations of classical planar quasiconformal maps for which the dilatation is allowed to tend to infinity in a controlled fashion. In this note we examine how these maps distort Hausdorff dimension. We show
- Given $\alpha$ and $\beta$ in $[0,2]$, there exists a David map $\varphi:\mathbb {C} \to \mathbb {C}$ and a compact set $\Lambda$ such that $\operatorname{dim}_{\operatorname{H}} \Lambda =\alpha$ and $\operatorname{dim}_{\operatorname{H}} \varphi(\Lambda)=\beta$.
- There exists a David map $\varphi:\mathbb {C} \to \mathbb {C}$ such that the Jordan curve $\Gamma=\varphi ({\mathbb S}^1)$ satisfies $\operatorname{dim}_{\operatorname{H}} \Gamma=2$.
One should contrast the first statement with the fact that quasiconformal maps preserve sets of Hausdorff dimension $0$ and $2$. The second statement provides an example of a Jordan curve with Hausdorff dimension $2$ which is (quasi)conformally removable.
The goal of this technical note is to show that the geometry of generalized parabolic towers cannot be essentially bounded. It fills a gap in author's paper "Combinatorics, geomerty and attractors of quasi-quadratic maps", Annals of Math., 1992.
This paper presents evidence for a conjecture concerning the structure of the set of braid types of periodic orbits of Smale's horseshoe map, partially ordered by Boyland's forcing order. The braid types are partitioned into totally ordered subsets, which are defined by parsing the symbolic code of a periodic orbit into two segments, the prefix and the decoration: the set of braid types of orbits with each given decoration is totally ordered, the order being given by the unimodal order on symbol sequences. The conjecture is supported by computer experiment, by proofs of special cases, and by intuitive argument in terms of pruning theory.
We describe the order on the ratios that define the generic universal smooth period doubling Cantor set. We prove that this set of ratios forms itself a Cantor set, a Conjecture formulated by Coullet and Tresser in 1977. We also show that the two period doubling renormalization operators, acting on the codimension one space of period doubling maps, form an iterated function system whose limit set contains a Cantor set.
Parabolic renormalization of critical circle maps arises as a degenerate case of the usual renormalization when the periods of the renormalized maps become infinite. In the paper we give new proofs of the main renormalization conjectures for the parabolic case, which are notably simplier than those required in the usual case. The title of the paper refers to the attracting fixed point of the parabolic renormalization, whose existence we prov
In this work we continue the exploration of affine and hyperbolic laminations associated with rational maps, which were introduced in LM. Our main goal is to construct natural geometric measures on these laminations: transverse conformal measures on the affine laminations and harmonic measures on the hyperbolic laminations. The exponent $\delta$ of the transverse conformal measure does not exceed 2, and is related to the eigenvalue of the harmonic measure by the formula $\lambda=\delta(\delta-2)$. In the course of the construction we introduce a number of geometric objects on the laminations: the basic cohomology class of an affine lamination (an obstruction to flatness), leafwise and transverse conformal streams, the backward and forward Poincaré series and the associated critical exponents. We discuss their relations to the Busemann and the Anosov--Sinai cocycles, the curvature form, currents and transverse invariant measures, $\lambda$-harmonic functions, Patterson--Sullivan and Margulis measures, etc. We also prove that the dynamical laminations in question are never flat except for several explicit special cases (rational functions with parabolic Thurston orbifold).