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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 . As diverges in the moduli space of
polynomials, the surface collapses along its foliation to yield a
metrized simplicial tree , 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 . We show that 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 provide a counterpart, in the setting of
iterated rational maps, to the -trees that arise as limits of
hyperbolic manifolds.Comment: 60 page
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Foliations of Hilbert Modular Surfaces
The Hilbert modular surface is the moduli space of Abelian varieties A with real multiplication by a quadratic order of discriminant . The locus where A is a product of elliptic curves determines a finite union of algebraic curves . In this paper we show the lamination extends to an essentially unique foliation of by complex geodesics. The geometry of is related to Teichm¨uller theory, holomorphic motions, polygonal billiards and Latt`es rational maps. We show every leaf of is either closed or dense, and compute its holonomy. We also introduce refinements of the classical modular curves on , leading to an explicit description of .Mathematic
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