John H. Hubbard

A proof of Thurston's topological characterization of rational functions

The criterion proved in this paper was stated by Thurston in November 1982. Thurston lectured on its proof on several occasions, notably at the NSF summer conference in Duluth, 1983, where one of the authors (JHH) was present. Using the notes of various attendants at these lectures, we have reconstructed a proof that we have made as precise as we could. Since this required a certain amount of work on our part, we thought it might be of some use to present this proof to the reader. We thank Dennis Sullivan for useful conversations, and Fritz yon Haesseler and especially Ben Wittner for help with the writing and valuable suggestions. After the first version was written, Clifford Earle pointed out that better estimates than what we had were to be found in [B].

Hénon mappings in the complex domain I : the global topology of dynamical space

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Hénon Mappings in the Complex Domain

Let H: ℂ2 → ℂ2 be the Henon mapping given by

THE HENON MAPPING IN THE COMPLEX DOMAIN

\left[ {_y^x} \right] \mapsto \left[ {_x^{p(x) - ay}} \right]

Groups of automorphisms of trees and their limit sets

The key invariant subsets are K ±, the sets of points with bounded forward images, J ± = ∂K ±, their boundaries, J = J + ∩ J −, and K = K + ∩ K −. In this paper we identify the topological structure of these sets when p is hyperbolic and |a| is sufficiently small, i.e., when H is a small perturbation of the polynomial p. The description involves projective and inductive limits of objects defined in terms of p alone.

On the existence and uniqueness of Strebel differentials

In 1969, P. Deligne and D. Mumford compactified the moduli space of curves. Their compactification is a projective algebraic variety, and as such, it has an underlying analytic structure. Alternatively, the quotient of the augmented Teichmueller space by the action of the mapping class group gives a compactification of the moduli space. We put an analytic structure on this compact quotient and prove that with respect to this structure, it is canonically isomorphic (as an analytic space) to the Deligne-Mumford compactification.

A First Look at Differential Algebra

Publisher Summary This chapter discusses Henon mapping in the complex domain. There is nothing about real polynomials that is independent of the coefficients, largely because virtually all features independent of conjugation such as periodic cycles are liable to disappear as the parameters are varied. In the complex domain, the behavior is far more uniform. The boundary of Fatou–Bieberbach domain is a topological manifold, and there are infinitely many analytic embeddings into the boundary whose images are dense. The chapter describes the rates of escape for the Henon family and a program for describing mappings in the Henon family.

The KAM Theorem

In this paper, a geometric interpretation is provided of a new rational Landen transformation. The convergence of its iterates is also established.