Douady-Earle section, holomorphic motions, and some applications

Proceedings of the American Mathematical Society, 1991

A holomorphic motion of £ c C over the unit disc D is a map /: DxC -> C such that f(0, w) = w , w € E , the function f{z, w) = fz(w) is holomorphic in z , and fz : E -» C is an injection for all z € D . Answering a question posed by Sullivan and Thurston [13], we show that every such / can be extended to a holomorphic motion F: D x C ->C. As a main step a "holomorphic axiom of choice" is obtained (concerning selections from the sets C\fz(E), z e D). The proof uses earlier results on the existence of analytic discs in the polynomial hulls of some subsets of C . Holomorphic motions are isotopies depending ho) omorphically on a complex parameter. Their study was originated by Mané et al. [8], in the context of the dynamics of rational maps, and was continued by Sullivan and Thurston and Bers and Royden . Definition . Let £ be a subset of C. A holomorphic motion of E in C, parametrized by the unit disc D, is a map f:DxE-*C such that (a) for any fixed w G E, the map z ^ fi(z, w): D ->C is holomorphic; (b) for any fixed z G D, the map w -► f(z, w) = fiz(w) is one-to-one; and (c) f0 is the identity map on X. Note that no continuity in w or (z, w) is assumed here. However, it holds due to the following remarkable "lambda lemma" of Mané et al. Lemma 1.1 [8]. If fi: D x E -> C is a holomorphic motion, then f(z, w) is jointly continuous and has a continuous extension to F: D x E -► C. Furthermore, F is a holomorphic motion of E over D, and the injections Fz(-) = F(z,-) are quasiconformal.

Contemporary Mathematics, 2012

We use Earle's generalization of Montel's theorem to obtain some results on holomorphic motions over infinite dimensional parameter spaces. We also study some properties of group-equivariant extensions of holomorphic motions.

Annales Academiae Scientiarum Fennicae Mathematica, 2012

We study some relationships between holomorphic motions, continuous motions, and monodromy. We also study extensions of holomorphic motions over Riemann surfaces and characterize the extendability of holomorphic motions over some planar regions in terms of monodromy.

arXiv (Cornell University), 2010

In this article we give an expository account of the holomorphic motion theorem based on work of Mãne-Sad-Sullivan, Bers-Royden, and Chirka. After proving this theorem, we show that tangent vectors to holomorphic motions have |ǫ log ǫ| moduli of continuity and then show how this type of continuity for tangent vectors can be combined with Schwarz's lemma and integration over the holomorphic variable to produce Hölder continuity on the mappings. We also prove, by using holomorphic motions, that Kobayashi's and Teichmüller's metrics on the Teichmüller space of a Riemann surface coincide. Finally, we present an application of holomorphic motions to complex dynamics, that is, we prove the Fatou linearization theorem for parabolic germs by involving holomorphic motions.

The Michigan Mathematical Journal, 2001

Transactions of the American Mathematical Society, 1994

We prove an equivariant form of Slodkowski's theorem that every holomorphic motion of a subset of the extended complex plane C extends to a holomorphic motion of C. As a consequence we prove that every holomorphic map of the unit disc into Teichmüller space lifts to a holomorphic map into the space of Beltrami forms. We use this lifting theorem to study the Teichmüller metric.

Nonlinearity, 2017

We study quasiconformal deformations and mixing properties of hyperbolic sets in the family of holomorphic correspondences z r + c, where r > 1 is rational. Julia sets in this family are projections of Julia sets of holomorphic maps on C 2 , which are skew-products when r is integer, and solenoids when r is non-integer and c is close to zero. Every hyperbolic Julia set in C 2 moves holomorphically. The projection determines a branched holomorphic motion with local (and sometimes global) parameterisations of the plane Julia set by quasiconformal curves.

The subject of holomorphic motions over the open unit disc has found important applications in complex dynamics. In this paper, we study holomorphic motions over more general parameter spaces. The Teichmiiller space of a closed subset of the Riemann sphere is shown to be a universal parameter space for holomorphic motions of the set over a simply connected complex Banach manifold. As a consequence, we prove a generalization of the "Harmonic A-Lemma" of Bers and Royden. We also study some other applications.

Annales Academiae Scientiarum Fennicae Mathematica, 2015

We develop an isotopy principle for holomorphic motions. Our main result concerns the extendability of a holomorphic motion of a finite subset E of a Riemann surface Y parameterized by a point t in a pointed hyperbolic surface (X, t0). If a holomorphic motion from E to Et in Y has a guiding quasiconformal isotopy, then there is a holomorphic extension to any new point p in Y − E that follows the guiding isotopy. The proof gives a canonical way to replace a quasiconformal motion of the (n + 1) − st point by a holomorphic motion while leaving unchanged the given holomorphic motion of the first n points. In particular, our main result gives a new proof of Slodkowski's theorem which concerns the special case when the parameter space is the open unit disk with base point 0 and the dynamical space Y is the Riemann sphere.