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Let $F\colon\mathbb{R}^2 \to \mathbb{R}^2$ be a homeomorphism. An open $F$-invariant subset $U$ of $\mathbb{R}^2$ is a pruning region for $F$ if it is possible to deform $F$ continuously to a homeomorphism $F_U$ for which every point of $U$ is wandering, but which has the same dynamics as $F$ outside of $U$. This concept was motivated by the {\em Pruning Front Conjecture} of Cvitanović, Gunaratne, and Procaccia, which claims that every Hénon map can be understood as a pruned horseshoe. This paper is a survey of pruning theory, concentrating on prunings of the horseshoe. We describe conditions on a disk $D$ which ensure that the orbit of its interior is a pruning region; explain how prunings of the horseshoe can be understood in terms of underlying tree maps; discuss the connection between pruning and Thurston's classification theorem for surface homeomorphisms; motivate a conjecture describing the forcing relation on horseshoe braid types; and use this theory to give a precise statement of the pruning front conjecture.
In this work we deal with non-discrete subgroups of ${\rm Diff}^{\omega} ({\mathbb S}^1)$, the group of orientation-preserving analytic diffeomorphisms of the circle. If $\Gamma$ is such a group, we consider its natural diagonal action ${\widetilde{\Gamma}}$ on the $n-$dimensional torus ${\mathbb T}^n$. It is then obtained a complete characterization of these groups $\Gamma$ whose corresponding ${\widetilde{\Gamma}}-$action on ${\mathbb T}^n$ is not piecewise ergodic (cf. Introduction) for all $n \in {\mathbb N}$ (cf. Theorem A). Theorem A can also be interpreted as an extension of Lie's classification of Lie algebras on ${\mathbb S}^1$ to general non-discrete subgroups of ${\mathbb S}^1$.
Given an orientation-preserving circle homeomorphism $h$, let $(E, \mathcal{L})$ denote a Thurston's left or right earthquake representation of $h$ and $\sigma $ the transversal shearing measure induced by $(E, \mathcal{L})$. We first show that the Thurston norm $||\cdot ||_{Th}$ of $\sigma $ is equivalent to the cross-ratio distortion norm $||\cdot ||_{cr}$ of $h$, i.e., there exists a constant $C>0$ such that $$\frac{1}{C}||h||_{cr}\le ||\sigma ||_{Th} \le C||h||_{cr}$$ for any $h$. Secondly we introduce two new norms on the cross-ratio distortion of $h$ and show they are equivalent to the Thurston norms of the measures of the left and right earthquakes of $h$. Together it concludes that the Thurston norms of the measures of the left and right earthquakes of $h$ and the three norms on the cross-ratio distortion of $h$ are all equivalent. Furthermore, we give necessary and sufficient conditions for the measures of the left and right earthquakes to vanish in different orders near the boundary of the hyperbolic plane. Vanishing conditions on either measure imply that the homeomorphism $h$ belongs to certain classes of circle diffeomorphisms classified by Sullivan in Sullivan.
The first two parts of this paper concern homeomorphisms of the circle, their associated earthquakes, earthquake laminations and shearing measures. We prove a finite version of Thurston's earthquake theorem Thurston4 and show that it implies the existence of an earthquake realizing any homeomorphism. Our approach gives an effective way to compute the lamination. We then show how to recover the earthquake from the measure, and give examples to show that locally finite measures on given laminations do not necessarily yield homeomorphisms. One of them also presents an example of a lamination $\mathcal {L}$ and a measure $\sigma $ such that the corresponding mapping $h_{\sigma}$ is not a homeomorphism of the circle but $h_{2\sigma}$ is.
The third part of the paper concerns the dependence between the norm $||\sigma ||_{Th}$ of a measure $\sigma$ and the norm $||h||_{cr}$ of its corresponding quasisymmetric circle homeomorphism $h_{\sigma}$. We first show that $||\sigma ||_{Th}$ is bounded by a constant multiple of $||h||_{cr}$. Conversely, we show for any $C_0>0$, there exists a constant $C>0$ depending on $C_0$ such that for any $\sigma $, if $||\sigma ||_{Th}\le C_0$ then $||h||_{cr}\le C||\sigma ||_{Th}$.
The fourth part of the paper concerns the differentiability of the earthquake curve $h_{t\sigma }, t\ge 0,$ on the parameter $t$. We show that for any locally finite measure $\sigma $, $h_{t\sigma }$ satisfies the nonautonomous ordinary differential equation $$\frac{d}{dt} h_{t\sigma}(x)=V_t(h_{t\sigma}(x)), \ t\ge 0,$$ at any point $x$ on the boundary of a stratum of the lamination corresponding to the measure $\sigma.$ We also show that if the norm of $\sigma $ is finite, then the differential equation extends to every point $x$ on the boundary circle, and the solution to the differential equation an initial condition is unique.
The fifth and last part of the paper concerns correspondence of regularity conditions on the measure $\sigma,$ on its corresponding mapping $h_{\sigma},$ and on the tangent vector $$V= V_0 = \frac{d}{dt}\big|_{t=0} h_{t\sigma}.$$ We give equivalent conditions on $\sigma, h_{\sigma}$ and $V$ that correspond to $h_{\sigma }$ being in Diff$^ {\ 1+\alpha}$ classes, where $0\le \alpha < 1$.
This paper gives a proof of the conjectural phenomena on the complex boundary one-dimensional slices: Every rational boundary point is cusp shaped. This paper treats this problem for Bers slices, the Earle slices, and the Maskit slice. In proving this, we also obtain the following result: Every Teichmüller modular transformation acting on a Bers slice can be extended as a quasi-conformal mapping on its ambient space. We will observe some similarity phenomena on the boundary of Bers slices, and discuss on the dictionary between Kleinian groups and Rational maps concerning with these phenomena. We will also give a result related to the theory of L.Keen and C.Series of pleated varieties in quasifuchsian space of once punctured tori.
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).
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
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.
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.
In this paper we extend M. Lyubich's recent results on the global hyperbolicity of renormalization of quadratic-like germs to the space $\mathbb{U}^r$ of $C^r$ unimodal maps with quadratic critical point. We show that in $\mathbb{U}^r$ the bounded-type limit sets of the renormalization operator have an invariant hyperbolic structure provided $r \ge 2+\alpha$ with $\alpha$ close to one. As an intermediate step between Lyubich's results and ours, we prove that the renormalization operator is hyperbolic in a Banach space of real analytic maps. We construct the local stable manifolds and prove that they form a continuous lamination whose leaves are $C^1$ codimension one Banach submanifolds of $\mathbb{U}^r$, and whose holonomy is $C^{1+\beta}$ for some $\beta>0$. We also prove that the global stable sets are $C^1$ immersed (codimension one) submanifolds as well, provided $r \ge 3+\alpha$ with $\alpha$ close to one. As a corollary, we deduce that in generic one parameter families of $C^r$ unimodal maps, the set of parameters corresponding to infinitely renormalizable maps of bounded combinatorial type is a Cantor set with Hausdorff dimension less than one.