Kevin M. Pilgrim

Combinations of complex dynamical systems

Introduction.- Preliminaries.- Combinations.- Uniqueness of combinations.- Decompositions.- Uniqueness of decompositions.- Counting classes of annulus maps.- Applications to mapping class groups. Examples.- Canonical decomposition theorem.

Quasisymmetrically inequivalent hyperbolic Julia sets.

We give explicit examples of pairs of Julia sets of hyperbolic rational maps which are homeomorphic but not quasisymmetrically homeomorphic.

Nearly Euclidean Thurston maps

We introduce and study a class of Thurston maps from the 2-sphere to itself which we call nearly Euclidean Thurston (NET) maps. These are simple generalizations of Euclidean Thurston maps.

Finite type coarse expanding conformal dynamics

We continue the study of non-invertible topological dynamical systems with expanding behavior. We introduce the class of {\em finite type} systems which are characterized by the condition that, up to rescaling and uniformly bounded distortion, there are only finitely many iterates. We show that subhyperbolic rational maps and finite subdivision rules (in the sense of Cannon, Floyd, Kenyon, and Parry) with bounded valence and mesh going to zero are of finite type. In addition, we show that the limit dynamical system associated to a selfsimilar, contracting, recurrent, level-transitive group action (in the sense of V. Nekrashevych) is of finite type. The proof makes essential use of an analog of the finiteness of cone types property enjoyed by hyperbolic groups.

An algebraic formulation of Thurston's combinatorial equivalence

Let f : S 2 → S 2 be an orientation-preserving branched covering for which the set P f of strict forward orbits of critical points is finite and let G = π 1 (S 2 -f -1 P f ). To f we associate an injective endomorphism φ f of the free group G, well-defined up to postcomposition with inner automorphisms. We show that two such maps f, g are combinatorially equivalent (in the sense introduced by Thurston for the characterization of rational functions as dynamical systems) if and only if Φ f , φ g are conjugate by an element of Out(G) which is induced by an orientation-preserving homeomorphism.

Spinning deformations of rational maps

We analyze a real one-parameter family of quasiconformal deformations of a hyperbolic rational map known as 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 Teichmuller theory is presented as well.

On the classification of critically fixed rational maps

We discuss the dynamical, topological, and algebraic classification of rational maps

Remarks on the period three cycles of quadratic rational maps

f

Origami, Affine Maps, and Complex Dynamics

of the Riemann sphere to itself each of whose critical points

Smale’s Mean Value Conjecture and Complex Dynamics

c