Harmonic mappings in the plane are univalent complex-valued harmonic functions of a complex variable. Conformal mappings are a special case where the real and imaginary parts are conjugate harmonic functions, satisfying the Cauchy-Riemann equations. Harmonic mappings were studied classically by differential geometers because they provide isothermal (or conformal) parameters for minimal surfaces. More recently they have been actively investigated by complex analysts as generalizations of univalent analytic functions, or conformal mappings. Many classical results of geometric function theory extend to harmonic mappings, but basic questions remain unresolved. This book is the first comprehensive account of the theory of planar harmonic mappings, treating both the generalizations of univalent analytic functions and the connections with minimal surfaces.  Essentially self-contained, the book contains background material in complex analysis and a full development of the classical theory of minimal surfaces, including the Weierstrass-Enneper representation. It is designed to introduce non-specialists to a beautiful area of complex analysis and geometry.

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Harmonic Mappings in the Plane

Let f(z) = ez+ z, where θ is an irrational number of bounded type. According to Siegel, f is linearizable on a disk containing the origin. In this paper we show: • the Hausdorff dimension of the Julia set J(f) is strictly less than two; and • if θ is a quadratic irrational (such as the golden mean), then the Siegel disk for f is self-similar about the critical point. In the latter case, we also show the rescaled first-return maps converge exponentially fast to a system of commuting branched coverings of the complex plane.

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Self-similarity of Siegel disks and Hausdorff dimension of Julia sets

Preface Introduction 1. Quasiconformal geometry 2. Extensions and interpolations 3. Preliminaries on dynamical systems and actions of Kleinian groups 4. Introduction to surgery and first occurrences 5. General principles of surgery 6. Soft surgeries with a contribution by X. Buff and C. Henriksen 7. Cut and paste surgeries with contributions by K. M. Pilgrim, Tan Lei and S. Bullett 8. Cut and paste surgeries with sectors with a contribution by A. L. Epstein and M. Yampolsky 9. Trans-quasiconformal surgery with contributions by C. L. Petersen and P. Haissinsky Bibliography Symbol index Index.

Quasiconformal surgery in holomorphic dynamics

Extending a result of Khavinson and Swiatek (2003) we show that the rational harmonic function r(z) - z, where r(z) is a rational function of degree n > 1, has no more than 5n - 5 complex zeros. Applications to gravitational lensing are discussed. In particular, this result settles a conjecture by Rhie concerning the maximum number of lensed images due to an n-point gravitational lens.

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On the number of zeros of certain rational harmonic functions

Let 0 << 1 be an irrational number with continued fraction expan- sion =( a1 ;a 2 ;a 3 ;::: ), and consider the quadratic polynomial P : z 7! e 2i z +z 2 . By performing a trans-quasiconformal surgery on an associated Blaschke product model, we prove that if log an = O( p n )a sn !1 ; then the Julia set of P is locally-connected and has Lebesgue measure zero. In particular, it follows that for almost every 0 << 1, the quadratic P has a Siegel disk whose boundary is a Jordan curve passing through the critical point of P.B y standard renormalization theory, these results generalize to the quadratics which have Siegel disks of higher periods.

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On the Julia set of a typical quadratic polynomial with a Siegel disk

Using techinques of complex dynamics we prove the conjecture of Sheil-Small and Wilmshurst that the harmonic polynomial z -p(z), degp = n > 1, has at most 3n - 2 complex zeros.

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On the number of zeros of certain harmonic polynomials

We prove that an analytic circle homeomorphism without periodic orbits is conjugated to the linear rotation by a quasi-symmetric map if an only if its rotation number is of constant type. Next, we consider automorphisms of quasi-conformal Jordan curves, without periodic orbits and holomorphic in a neighborhood. We prove a “Denjoy theorem” that such maps are conjugated to a rotation on the circle.

On critical circle homeomorphisms

Resume L'application de premier retour d'une application unimodale C 3 dans un voisinage du point critique est conjuguee par un diffeomorphisme reel-analytique a une application dont la derivee schwarzienne est negative. Comme consequence, on obtient la classification des attracteurs metriques pour les applications unimodales lisses avec point critique non degenere.

La dérivée schwarzienne en dynamique unimodale

We consider infinitely renormalizable unimodal mappings with topological type which are periodic under renormalization. We study the limiting behavior of fixed points of the renormalization operator as the order of the critical point increases to infinity. It is shown that a limiting dynamics exists, with a critical point that is flat, but still having a well-behaved analytic continuation to a neighborhood of the real interval pinched at the critical point. We study the dynamics of limiting maps and prove their rigidity. In particular, the sequence of fixed points of renormalization for finite criticalities converges, uniformly on the real domain, to a mapping of the limiting type.

Dynamics and universality of unimodal mappings with infinite criticality

We consider unimodal polynomials with Feigenbaum topological type and critical points whose orders tend to infinity. It is shown that the hyperbolic dimensions of their Julia set go to 2; furthermore, that the Hausdorff dimensions of the basins of attraction of their Feigenbaum attractors also tend to 2. The proof is based on constructing a limiting dynamics with a flat critical point.

Grzegorz Światek

Papers

On the number of zeros of certain harmonic polynomials

On critical circle homeomorphisms

La dérivée schwarzienne en dynamique unimodale

Dynamics and universality of unimodal mappings with infinite criticality

Hausdorff Dimension of Julia Sets of Feigenbaum Polynomials with High Criticality