Teichm¨ullertheory and dynamical systems

Jeremy Kahn Fall 2013–Spring 2014 May 17, 2013

The intention of this year-long seminar course is to cover four subjects: an introduction to rational maps and Kleinian groups, an introduction to Teichm¨ullertheory and the Teichm¨ullermetric, the “classical” applications of Teichm¨ullertheory to rational maps and Kleinian groups, and my own work, in part with Mikhail Lyubich, on the asymptotic theory of moduli and its applications to conformal dynamical systems. Prerequisites are the first year courses in real and complex analysis, or the equivalent. Not everything listed in the course description will necessarily be covered.

Kleinian groups and rational maps A rational map is a holomorphic map from the Riemann sphere to itself; a Kleinian group is a discrete group of holomorphic automorphisms of the Rie- mann sphere. Both of these might be called conformal dynamical systems. In this first part we will define the Julia set of a rational map[Mil06] and the limit set of a Kleinian group, and then describe the dynamics on the com- plementary components of the rational map/limit set. We will introduce the Sullivan dictionary between rational maps and Kleinian groups, and describe a number of classical results and problems, such as the Ending Lamination Conjecture[Min10, BCM12] and the Local Connectivity of the Mandelbrot Set.

1 Teichm¨ullerTheory The Teichm¨ullerspace of a Riemann surface is the space of all deformations of the complex structure, with a topological marking that insures that the Teichm¨ullerspace is simply connected. We will define a quasiconformal map and use this to define the Teichmuller metric on Teichmuller space[Ahl06, Gar87, Hub06]. We will then introduce the Bers embedding and the theory of extremal length and use these to prove the Teichm¨ullerexistence and uniqueness theorem, which describes the geodesics in Teichm¨ullerspace. We will also describe how the integrable holomorphic quadratic differentials serve as the cotangent space to Teichmuller space.

Teichm¨ullerTheory and conformal dynamical systems We will describe the great classical applications of Teichm¨ullertheory to the theory of rational maps and Kleinian groups:

1. The Ahlfors Finiteness Theorem and Sullivan’s No Wandering Domains[Sul85]

2. The Classification of Postcritically Finite Rational Maps[DH93]

3. The Classification of elements of the Mapping Class Group

4. The deformation theory for geometrically finite rational maps and Kleinian groups (including Mostow rigidity)[MS98]

5. The Theta conjecture and the contraction of the skinning map[McM89, McM90].

The theory of degenerate complex structures We will introduce the problem of the asymptotic theory of moduli, where we ask how ratios of moduli of path families behave as the moduli go to infinity. We show in many cases how this can be answered with the theory of degenerate complex structures, and how this answer can be applied to prove classical conjectures about rational maps. We begin with two simple compactifications of universal Teichm¨uller space, one with laminations and one with measured laminations. We describe how the second one leads to Thurston’s compactification of the Teichm¨uller space of a surface with the space of projective measured laminations. Then

2 we introduce the theory of modular laminations and use it to prove the Cov- ering Lemma and the Quasi-Additivity Law[KL09a]. We will then describe how these theorems relate to the progress made toward the bounds for renor- malization and the Local Connectivity of the Mandelbrot set[KL09b, KL08].

References

[Ahl06] Lars V. Ahlfors. Lectures on quasiconformal mappings, volume 38 of University Lecture Series. American Mathematical Society, Providence, RI, second edition, 2006. With supplemental chap- ters by C. J. Earle, I. Kra, M. Shishikura and J. H. Hubbard.

[BCM12] Jeffrey F. Brock, Richard D. Canary, and Yair N. Minsky. The classification of Kleinian surface groups, II: The ending lamination conjecture. Ann. of Math. (2), 176(1):1–149, 2012.

[DH93] Adrien Douady and John H. Hubbard. A proof of Thurston’s topological characterization of rational functions. Acta Math., 171(2):263–297, 1993.

[Gar87] Frederick P. Gardiner. Teichm¨uller theory and quadratic differen- tials. Pure and Applied Mathematics (New York). John Wiley & Sons Inc., New York, 1987. A Wiley-Interscience Publication.

[Hub06] John Hamal Hubbard. Teichm¨uller theory and applications to ge- ometry, topology, and dynamics. Vol. 1. Matrix Editions, Ithaca, NY, 2006. Teichm¨ullertheory, With contributions by Adrien Douady, William Dunbar, Roland Roeder, Sylvain Bonnot, David Brown, Allen Hatcher, Chris Hruska and Sudeb Mitra, With fore- words by William Thurston and Clifford Earle.

[KL08] Jeremy Kahn and Mikhail Lyubich. A priori bounds for some infinitely renormalizable quadratics. II. Decorations. Ann. Sci. Ec.´ Norm. Sup´er.(4), 41(1):57–84, 2008.

[KL09a] Jeremy Kahn and Mikahil Lyubich. The quasi-additivity law in conformal geometry. Ann. of Math. (2), 169(2):561–593, 2009.

3 [KL09b] Jeremy Kahn and Mikhail Lyubich. Local connectivity of Julia sets for unicritical polynomials. Ann. of Math. (2), 170(1):413– 426, 2009.

[McM89] Curt McMullen. Amenability, Poincar´eseries and quasiconformal maps. Invent. Math., 97(1):95–127, 1989.

[McM90] C. McMullen. Iteration on Teichm¨ullerspace. Invent. Math., 99(2):425–454, 1990.

[Mil06] John Milnor. Dynamics in one complex variable, volume 160 of An- nals of Mathematics Studies. Princeton University Press, Prince- ton, NJ, third edition, 2006.

[Min10] Yair Minsky. The classification of Kleinian surface groups. I. Mod- els and bounds. Ann. of Math. (2), 171(1):1–107, 2010.

[MS98] Curtis T. McMullen and Dennis P. Sullivan. Quasiconformal home- omorphisms and dynamics. III. The Teichm¨ullerspace of a holo- morphic dynamical system. Adv. Math., 135(2):351–395, 1998.

[Sul85] Dennis Sullivan. Quasiconformal homeomorphisms and dynamics. I. Solution of the Fatou-Julia problem on wandering domains. Ann. of Math. (2), 122(3):401–418, 1985.

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