Frontiers in complex dynamics

Rational maps on the Riemann sphere occupy a distinguished niche in the general theory of smooth dynamical systems. First, rational maps are complex-analytic, so a broad spectrum of techniques can contribute to their study (quasiconformal mappings, potential theory, algebraic geometry, etc.). The rational maps of a given degree form a finite-dimensional manifold, so exploration of this {\em parameter space} is especially tractable. Finally, some of the conjectures once proposed for {\em smooth} dynamical systems (and now known to be false) seem to have a definite chance of holding in the arena of rational maps. In this article we survey a small constellation of such conjectures centering around the density of {\em hyperbolic} rational maps --- those which are dynamically the best behaved. We discuss some of the evidence and logic underlying these conjectures, and sketch recent progress towards their resolution.Comment: 18 pages. Abstract added in migration

Trees and the dynamics of polynomials

The basin of infinity of a polynomial map f : {\bf C} \arrow {\bf C} carries a natural foliation and a flat metric with singularities, making it into a metrized Riemann surface $X(f)$. As $f$ diverges in the moduli space of polynomials, the surface $X(f)$ collapses along its foliation to yield a metrized simplicial tree $(T,\eta)$, with limiting dynamics F : T \arrow T. In this paper we characterize the trees that arise as limits, and show they provide a natural boundary \PT_d compactifying the moduli space of polynomials of degree $d$. We show that $(T,\eta,F)$ records the limiting behavior of multipliers at periodic points, and that any divergent meromorphic family of polynomials \{f_t(z) : t \mem \Delta^* \} can be completed by a unique tree at its central fiber. Finally we show that in the cubic case, the boundary of moduli space \PT_3 is itself a tree. The metrized trees $(T,\eta,F)$ provide a counterpart, in the setting of iterated rational maps, to the ${\bf R}$-trees that arise as limits of hyperbolic manifolds.Comment: 60 page

Foliations of Hilbert Modular Surfaces

The Hilbert modular surface $X_D$ is the moduli space of Abelian varieties A with real multiplication by a quadratic order of discriminant $D > 1$. The locus where A is a product of elliptic curves determines a finite union of algebraic curves $X_D(1) \subset X_D$. In this paper we show the lamination $X_D(1)$ extends to an essentially unique foliation $F_D$ of $X_D$ by complex geodesics. The geometry of $F_D$ is related to Teichm¨uller theory, holomorphic motions, polygonal billiards and Latt`es rational maps. We show every leaf of $F_D$ is either closed or dense, and compute its holonomy. We also introduce refinements $T_N(\nu)$ of the classical modular curves on $X_D$, leading to an explicit description of $X_D(1)$.Mathematic

Teichmüller Geodesics of Infinite Complexity

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Dynamics on Algebraic Surfaces

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The Moduli Space of Riemann Surfaces is Kahler Hyperbolic

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Rational Maps and Teichmu¨llerTeichm\ddot{u}ller Space

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Hausdorff Dimension of General Sierpinski Carpets

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Coxeter Groups, Salem Numbers and the Hilbert Metric

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