Holomorphic Motions Research Articles

Holomorphic motions, dimension, area and quasiconformal mappings

We describe the variation of the Minkowski, packing and Hausdorff dimensions of a set moving under a holomorphic motion, as well as the variation of its area. Our method provides a new, unified approach to various celebrated theorems about quasiconformal mappings, including the work of Astala on the distortion of area and dimension under quasiconformal mappings and the work of Smirnov on the dimension of quasicircles.

Logarithmic capacity under holomorphic motions in higher dimensions

We show that logarithmic capacity in even dimensional Euclidean spaces varies subharmonically under holomorphic motions.

Analytic linearization of conformal maps of the annulus

We consider holomorphic maps defined in an annulus around R/Z in C/Z. E. Risler proved that in a generic analytic family of such maps fζ that contains a Brjuno rotation f0(z)=z+α, all maps that are conjugate to this rotation form a codimension-1 analytic submanifold near f0.In this paper, we obtain the Risler's result as a corollary of the following construction. We introduce a renormalization operator on the space of univalent maps in a neighborhood of R/Z. We prove that this operator is hyperbolic, with one unstable direction corresponding to translations. We further use a holomorphic motions argument and Yoccoz's theorem to show that its stable foliation consists of diffeomorphisms that are conjugate to rotations.

Holomorphic explosions — I

This paper concerns the problem of extending the parameter domain of holomorphic motions to include isolated boundary points, punctures of the domains. Supposing that the parameter domain has an isolated boundary point λ ∗ $\lambda ^*$ , we explore the extension properties of the holomorphic motion to λ ∗ $\lambda ^*$ .

Stretching and Rotation of Planar Quasiconformal Mappings on a Line

In this article, we examine stretching and rotation of planar quasiconformal mappings on a line. We show that for almost every point on the line, the set of complex stretching exponents (describing stretching and rotation jointly) is contained in the disk overline {B}(1/(1-k^{4}),k^{2}/(1-k^{4})). This yields a quadratic improvement over the known optimal estimate for general sets of Hausdorff dimension 1. Our proof is based on holomorphic motions and estimates for dimensions of quasicircles. We also give a lower bound for the dimension of the image of a 1-dimensional subset of a line under a quasiconformal mapping.

Dynamics of hyperbolic correspondences

AbstractThis paper establishes the geometric rigidity of certain holomorphic correspondences in the family $(w-c)^q=z^p$ , whose post-critical set is finite in any bounded domain of $\mathbb {C}$ . In spite of being rigid on the sphere, such correspondences are J-stable by means of holomorphic motions when viewed as maps of $\mathbb {C}^2$ . The key idea is the association of a conformal iterated function system to the return branches near the critical point, giving a global description of the post-critical set and proving the hyperbolicity of these correspondences.

Continuity of capacity of a holomorphic motion

We study the behavior of various set-functions under holomorphic motions. We show that, under such deformations, logarithmic capacity varies continuously, while analytic capacity may not.

Continuity of condenser capacity under holomorphic motions

AbstractWe show that condenser capacity varies continuously under holomorphic motions, and the corresponding family of the equilibrium measures of the condensers is continuous with respect to the weak-star convergence. We also study the behavior of uniformly perfect sets under holomorphic motions.

Positive transversality via transfer operators and holomorphic motions with applications to monotonicity for interval maps

In this paper we will develop a general approach which shows that generalized ‘critical relations’ of families of locally defined holomorphic maps on the complex plane unfold transversally. The main idea is to define a transfer operator, which is a local analogue of the Thurston pullback operator, using holomorphic motions. Assuming a so-called lifting property is satisfied, we obtain information about the spectrum of this transfer operator and thus about transversality. An important new feature of our method is that it is not global: the maps we consider are only required to be defined and holomorphic on a neighbourhood of some finite set. We will illustrate this method by obtaining transversality for a wide class of one-parameter families of interval and circle maps, for example for maps with flat critical points, but also for maps with complex analytic extensions such as certain polynomial-like maps. As in Tsujii’s approach (Tsujii M 1994 A note on Milnor and Thurston’s monotonicity theorem Geometry and Analysis in Dynamical System vol 14 (Singapore: World Scientific) pp 60–2; Tsujii M 2000 Ergod. Theor. Dyn. Syst. 20 925–933), for real maps we obtain positive transversality (where >0 holds instead of just ≠0), and thus monotonicity of entropy for these families, and also (as an easy application) for the real quadratic family. This method additionally gives results for unimodal families of the form x ↦ |x|ℓ + c for ℓ > 1 not necessarily an even integer and c real.

Winding numbers and full extendibility in holomorphic motions

We construct an example of a holomorphic motion of a five-point subset of the Riemann sphere over an annulus such that it satisfies the zero winding number condition but is not fully extendable.

A note on the extendability of holomorphic motions

In this paper, we consider conditions under which a holomorphic motion of a closed subset of $\mathbf{Ĉ}$ over a non-simply connected Riemann surface $X$ can be extended to a holomorphic motion of $\mathbf{Ĉ}$ over $X$. We construct examples of non-extendable holomorphic motions which satisfy fairy good topological conditions. The examples are also counter-examples to a claim by Chirka for the extendability of holomorphic motions.

Analytic capacity and holomorphic motions

We study the behavior of the analytic capacity of a compact set under deformations obtained by families of conformal maps depending holomorphically on the complex parameter. We show that, under those deformations, the logarithm of the analytic capacity varies harmonically. We also show that the hypotheses in this result cannot be substantially weakened.

Monodromy, liftings of holomorphic maps, and extensions of holomorphic motions

We study monodromy of holomorphic motions and show the equivalence of triviality of monodromy of holomorphic motions and extensions of holomorphic motions to continuous motions of the Riemann sphere. We also study liftings of holomorphic maps into certain Teichmüller spaces. We use this “lifting property” to prove that, under the condition of trivial monodromy, any holomorphic motion of a closed set in the Riemann sphere, over a hyperbolic Riemann surface, can be extended to a holomorphic motion of the sphere, over the same parameter space. We conclude that this extension can be done in a conformally natural way.

Cycle doubling, merging, and renormalization in the tangent family

In this paper we study the transition to chaos for the restriction to the real and imaginary axes of the tangent family { T t ( z ) = i t tan ⁡ z } 0 > t ≤ π \{ T_t(z)=i t\tan z\}_{0> t\leq \pi } . Because tangent maps have no critical points but have an essential singularity at infinity and two symmetric asymptotic values, there are new phenomena: as t t increases we find single instances of “period quadrupling”, “period splitting”, and standard “period doubling”; there follows a general pattern of “period merging” where two attracting cycles of period 2 n 2^n “merge” into one attracting cycle of period 2 n + 1 2^{n+1} , and “cycle doubling” where an attracting cycle of period 2 n + 1 2^{n+1} “becomes” two attracting cycles of the same period. We use renormalization to prove the existence of these bifurcation parameters. The uniqueness of the cycle doubling and cycle merging parameters is quite subtle and requires a new approach. To prove the cycle doubling and merging parameters are, indeed, unique, we apply the concept of “holomorphic motions” to our context. In addition, we prove that there is an “infinitely renormalizable” tangent map T t ∞ T_{t_\infty } . It has no attracting or parabolic cycles. Instead, it has a strange attractor contained in the real and imaginary axes which is forward invariant and minimal under T t ∞ 2 T^2_{t_\infty } . The intersection of this strange attractor with the real line consists of two binary Cantor sets and the intersection with the imaginary line is totally disconnected, perfect, and unbounded.

Holomorphic motions and complex geometry

We show that the graph of a holomorphic motion of the unit disc cannot be biholomorphic to a strongly pseudoconvex domain in C-n.

Families of Riemann surfaces, uniformization and arithmeticity

A consequence of the results of Bers and Griffiths on the uniformization of complex algebraic varieties is that the universal cover of a family of Riemann surfaces, with base and fibers of finite hyperbolic type, is a contractible 2 − 2- dimensional domain that can be realized as the graph of a holomorphic motion of the unit disk. In this paper we determine which holomorphic motions give rise to these uniformizing domains and characterize which among them correspond to arithmetic families (i.e. families defined over number fields). Then we apply these results to characterize the arithmeticity of complex surfaces of general type in terms of the biholomorphism class of the 2 − 2- dimensional domains that arise as universal covers of their Zariski open subsets. For the important class of Kodaira fibrations this criterion implies that arithmeticity can be read from the universal cover. All this is very much in contrast with the corresponding situation in complex dimension one, where the universal cover is always the unit disk.

Holomorphic motions for unicritical correspondences

We study quasiconformal deformations and mixing properties of hyperbolic sets in the family of holomorphic correspondences zr + c, where r > 1 is rational. Julia sets in this family are projections of Julia sets of holomorphic maps on 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 moves holomorphically. The projection determines a branched holomorphic motion with local (and sometimes global) parameterizations of the plane Julia set by quasiconformal curves.

Stream lines, quasilines and holomorphic motions

We give a new application of the theory of holomorphic motions to study the distortion of level lines of harmonic functions and stream lines of ideal planar fluid flow. In various settings, we show they are in fact quasilines—the quasiconformal images of the real line. These methods also provide quite explicit global estimates on the geometry of these curves.

Guiding isotopies and holomorphic motions

We develop an isotopy principle for holomorphic motions. Our main result concerns the extendability of a holomorphic motion h(t,z) of a finite subset E of the Riemann sphere C parameterized by a pointed hyperbolic Riemann surface (X,t0). We prove that if this holomorphic motion has a guiding quasiconformal isotopy, then it can be extended to a new holomorphic motion of E ∪ {p} for any point p in C E that follows the guiding isotopy. The proof gives a canonical way to replace a continuous motion of the (n + 1)-st point by a holomorphic motion while leaving unchanged the given holomorphic motion of the first n points.

Holomorphic motions and quasicircles

Holomorphic motions and quasicircles