Simple Closed Curves and Complex Dynamics

William Thurston’s Topological Characterization of Rational Maps is a central theorem in complex dynamics. It contains a topological criterion under which a branched cover S^2 -> S^2 is equivalent to a rational map on the Riemann sphere. This criterion is formulated in terms of simple closed curves on the sphere. We will explore this and discuss different kinds of behavior that these curve systems exhibit for branched covers and rational maps.

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