Let $f: S^2 \to S^2$ be an expanding branched covering map of the sphere to itself with finite postcritical set $P_f$. Associated to $f$ is a canonical quasisymmetry class $\GGG(f)$ of Ahlfors regular metrics on the sphere in which the... more
Let $f: S^2 \to S^2$ be an expanding branched covering map of the sphere to itself with finite postcritical set $P_f$. Associated to $f$ is a canonical quasisymmetry class $\GGG(f)$ of Ahlfors regular metrics on the sphere in which the dynamics is (non-classically) conformal. We show \[ \inf_{X \in \GGG(f)} \hdim(X) \geq Q(f)=\inf_\Gamma \{Q \geq 2: \lambda(f_{\Gamma,Q}) \geq 1\}.\] The infimum is over all multicurves $\Gamma \subset S^2-P_f$. The map $f_{\Gamma,Q}: \R^\Gamma \to \R^\Gamma$ is defined by \[ f_{\Gamma, Q}(\gamma) =\sum_{[\gamma']\in\Gamma} \sum_{\delta \sim \gamma'} \deg(f:\delta \to \gamma)^{1-Q}[\gamma'],\] where the second sum is over all preimages $\delta$ of $\gamma$ freely homotopic to $\gamma'$ in $S^2-P_f$, and $ \lambda(f_{\Gamma,Q})$ is its Perron-Frobenius leading eigenvalue. This generalizes Thurston's observation that if $Q(f)>2$, then there is no $f$-invariant classical conformal structure.
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We prove that if the Ahlfors regular conformal dimension $Q$ of a topologically cxc map on the sphere $f: S^2 \to S^2$ is realized by some metric $d$ on $S^2$, then either Q=2 and $f$ is topologically conjugate to a semihyperbolic... more
We prove that if the Ahlfors regular conformal dimension $Q$ of a topologically cxc map on the sphere $f: S^2 \to S^2$ is realized by some metric $d$ on $S^2$, then either Q=2 and $f$ is topologically conjugate to a semihyperbolic rational map with Julia set equal to the whole Riemann sphere, or $Q>2$ and $f$ is topologically conjugate to a map which lifts to an affine expanding map of a torus whose differential has distinct real eigenvalues. This is an analog of a known result for Gromov hyperbolic groups with two-sphere boundary, and our methods apply to give a new proof.
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Let $M_d$ be the moduli space of one-dimensional complex polynomial dynamical systems. The escape rates of the critical points determine a critical heights map $G: M_d \to \mathbb{R}^{d-1}$. For generic values of $G$, each connected... more
Let $M_d$ be the moduli space of one-dimensional complex polynomial dynamical systems. The escape rates of the critical points determine a critical heights map $G: M_d \to \mathbb{R}^{d-1}$. For generic values of $G$, each connected component of a fiber of $G$ is the deformation space for twist deformations on the basin of infinity. We analyze the quotient space $\mathcal{T}_d^*$ obtained
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We consider the problem of classifying the dynamics of complex polynomials $f: \mathbb{C} \to \mathbb{C}$ restricted to their basins of infinity. We synthesize existing combinatorial tools --- tableaux, trees, and laminations --- into a... more
We consider the problem of classifying the dynamics of complex polynomials $f: \mathbb{C} \to \mathbb{C}$ restricted to their basins of infinity. We synthesize existing combinatorial tools --- tableaux, trees, and laminations --- into a new invariant of basin dynamics we call the pictograph. For polynomials with all critical points escaping to infinity, we obtain a complete description of the set
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We study the projection $\pi: M_d \to B_d$ which sends an affine conjugacy class of polynomial $f: \mathbb{C}\to\mathbb{C}$ to the holomorphic conjugacy class of the restriction of $f$ to its basin of infinity. When $B_d$ is equipped with... more
We study the projection $\pi: M_d \to B_d$ which sends an affine conjugacy class of polynomial $f: \mathbb{C}\to\mathbb{C}$ to the holomorphic conjugacy class of the restriction of $f$ to its basin of infinity. When $B_d$ is equipped with a dynamically natural Gromov-Hausdorff topology, the map $\pi$ becomes continuous and a homeomorphism on the shift locus. Our main result is that
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We study dynamical equivalence relations on the moduli space MPd of complex polynomial dynamical systems. Our main result is that the critical-heights quotient MPd ! T d of (DP1) is the Hausdorzation of a relation based on the twisting... more
We study dynamical equivalence relations on the moduli space MPd of complex polynomial dynamical systems. Our main result is that the critical-heights quotient MPd ! T d of (DP1) is the Hausdorzation of a relation based on the twisting deformation of the basin of infinity. We also study relations of topological conjugacy and the Branner-Hubbard wringing deformation.
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Building on the dictionary between Kleinian groups and rational maps, we establish new connections between the theories of hyperbolic groups and certain iterated maps, regarded as dynamical systems. In order to make the exposition... more
Building on the dictionary between Kleinian groups and rational maps, we establish new connections between the theories of hyperbolic groups and certain iterated maps, regarded as dynamical systems. In order to make the exposition self-contained to researchers in many fields, we include detailed proofs and ample background.
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Let f : S2 ! S2 be a postcritically finite expanding branched covering map of the sphere to itself. Associated to f is a canonical quasisymmetry class G(f) of Ahlfors regular metrics on the sphere in which the dynamics is... more
Let f : S2 ! S2 be a postcritically finite expanding branched covering map of the sphere to itself. Associated to f is a canonical quasisymmetry class G(f) of Ahlfors regular metrics on the sphere in which the dynamics is (non-classically) conformal. We find a lower bound on the Hausdorff dimension of metrics in G(f) in terms of the combinatorics
We show that if f is a hyperbolic rational map with disconnected Julia set J , then with the possible exception of finitely many periodic components of J and their countable collection of preimages, every connected component of J is a... more
We show that if f is a hyperbolic rational map with disconnected Julia set J , then with the possible exception of finitely many periodic components of J and their countable collection of preimages, every connected component of J is a point or a Jordan curve. As a corollary, every component of J is locally connected. We also discuss when
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Page 1. Remarks on the period three cycles of quadratic rational maps This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2003 Nonlinearity 16 93... more
Page 1. Remarks on the period three cycles of quadratic rational maps This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2003 Nonlinearity 16 93 (http://iopscience.iop.org/0951-7715/16/1/306) ...
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Abstract. We discuss the general combinatorial, topological, algebraic, and dynamical issues underlying the enumeration of postcritically finite rational functions, regarded as holomorphic dynamical systems on the Riemann sphere. We... more
Abstract. We discuss the general combinatorial, topological, algebraic, and dynamical issues underlying the enumeration of postcritically finite rational functions, regarded as holomorphic dynamical systems on the Riemann sphere. We present findings from our ...
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ABSTRACT For n≥2, the family of rational maps F λ (z)=z n +λ/z n contains a countably infinite set of parameter values for which all critical orbits eventually land after some number κ of iterations on the point at infinity. The Julia... more
ABSTRACT For n≥2, the family of rational maps F λ (z)=z n +λ/z n contains a countably infinite set of parameter values for which all critical orbits eventually land after some number κ of iterations on the point at infinity. The Julia sets of such maps are Sierpiński curves if κ≥3. We show that two such maps are topologically conjugate on their Julia sets if and only if they are Möbius or anti-Möbius conjugate, and we give a precise count of the number of topological conjugacy classes as a function of n and κ.
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ABSTRACT
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We study dynamical equivalence relations on the moduli space MPd of complex polynomial dynamical systems. Our main result is that the critical-heights quotient MPd ! T d of (DP1) is the Hausdorzation of a relation based on the twisting... more
We study dynamical equivalence relations on the moduli space MPd of complex polynomial dynamical systems. Our main result is that the critical-heights quotient MPd ! T d of (DP1) is the Hausdorzation of a relation based on the twisting deformation of the basin of infinity. We also study relations of topological conjugacy and the Branner-Hubbard wringing deformation.
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We discuss the dynamical, topological, and algebraic classification of rational maps f : C → C, each of whose critical points c is also a fixed-point of f , i.e., f (c) = c.