Irreducibility in complex dynamics

Abstract: A major goal in complex dynamics is to understand dynamical moduli spaces; that is, conjugacy classes of holomorphic dynamical systems. One of the great successes in this regard is the study of the moduli space of quadratic polynomials; it is isomorphic to the complex plane. This moduli space contains the famous Mandelbrot set, which has been extensively studied over the past 40 years. Understanding other dynamical moduli spaces to the same extent tends to be more challenging as they are often higher-dimensional. In this talk, we consider the moduli space of quadratic rational maps, which is isomorphic to C^2. We will focus on special algebraic curves, called "Milnor curves" in this space. In general, it is unknown if Milnor curves are irreducible over C. Because these curves are smooth, this is equivalent to asking whether they are connected. We will exhibit the first infinite collection of Milnor curves that are connected. This is joint work with X. Buff and A. Epstein.

analysis of PDEscomplex variablesdynamical systemsmetric geometry

Audience: researchers in the topic

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