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Kevin M. Pilgrim

Publications and source records attributed to Kevin M. Pilgrim.

26 records · Page 2Linked to original sources
arXiv2011

Minimal Ahlfors regular conformal dimension of coarse conformal dynamics on the sphere

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.

math.DS↗
arXiv2010

An algebraic formulation of Thurston's characterization of rational functions

Following Douady-Hubbard and Bartholdi-Nekrashevych, we give an algebraic formulation of Thurston's characterization of rational functions. The techniques developed are applied to the analysis of the dynamics on the set of free homotopy classes of simple closed curves induced by a rational function. The resulting finiteness results yield new information on the global dynamics of the pullback map on Teichmüller space used in the proof of the characterization theorem.

math.DS↗
arXiv2010

Examples of coarse expanding conformal maps

In previous work, a class of noninvertible topological dynamical systems $f: X \to X$ was introduced and studied; we called these {\em topologically coarse expanding conformal} systems. To such a system is naturally associated a preferred quasisymmetry (indeed, snowflake) class of metrics in which arbitrary iterates distort roundness and ratios of diameters by controlled amounts; we called this {\em metrically coarse expanding conformal}. In this note we extend the class of examples to several more familiar settings, give applications of our general methods, and discuss implications for the computation of conformal dimension.

math.DS↗
arXiv2009

Coarse expanding conformal dynamics

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.

math.DS↗
arXiv2008

Thurston obstructions and Ahlfors regular conformal dimension

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 \[ \inf_{X \in \GGG(f)} \hdim(X) \geq Q(f)=\inf_Γ\{Q \geq 2: λ(f_{Γ,Q}) \geq 1\}.\] The infimum is over all multicurves $Γ\subset S^2-P_f$. The map $f_{Γ,Q}: \R^Γ\to \R^Γ$ is defined by \[ f_{Γ, Q}(γ) =\sum_{[γ']\inΓ} \sum_{δ\sim γ'} °(f:δ\to γ)^{1-Q}[γ'],\] where the second sum is over all preimages $δ$ of $γ$ freely homotopic to $γ'$ in $S^2-P_f$, and $ λ(f_{Γ,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.

math.DS↗
arXiv2002

Spinning deformations of rational maps

We analyze a real one-parameter family of quasiconformal deformations of a hyperbolic rational map known as {\em spinning}. We show that under fairly general hypotheses, the limit of spinning either exists and is unique, or else converges to infinity in the moduli space of rational maps of a fixed degree. When the limit exists, it has either a parabolic fixed point, or a pre-periodic critical point in the Julia set, depending on the combinatorics of the data defining the deformation. The proofs are soft and rely on two ingredients: the construction of a Riemann surface containing the closure of the family, and an analysis of the geometric limits of some simple dynamical systems. An interpretation in terms of Teichmüller theory is presented as well.

math.DS↗
arXiv1994

Rational Maps Whose Fatou Components Are Jordan Domains

We prove: If $f(z)$ is a critically finite rational map which has exactly two critical points and which is not conjugate to a polynomial, then the boundary of every Fatou component of $f$ is a Jordan curve. If $f(z)$ is a hyperbolic critically finite rational map all of whose postcritical points are periodic, then there exists a cycle of Fatou components whose boundaries are Jordan curves. We give examples of critically finite hyperbolic rational maps $f$ with the property that on the closure of a Fatou component $Ω$ satisfying $f(Ω)=Ω$, $f|_{\bdry Ω}$ is not topologically conjugate to the dynamics of any polynomial on its Julia set.

math.DS↗