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Special Colloquium
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Rohini Ramadas, Zoom: Dynamics on the moduli space of point-configurations on the Riemann sphere
Tuesday, January 12, 2021, 01:00pm - 02:00pm
A rational function f(z) in one variable with complex coefficients defines a holomorphic self-map of the Riemann sphere. The field of complex dynamics was born around the beginning of the 20th century with the study of iteration of rational functions. A degree-d rational function f(z) has (at most) 2d-2 critical points, i.e. points at which the derivative f'(z) vanishes. The trajectories of the finitely many critical points of f(z) under iteration strongly influence the global dynamics of f(z). A rational function is called post-critically finite (PCF) if every critical point is (pre)-periodic. PCF rational functions have been central in complex dynamics, due to their special dynamical behavior, and their special distribution within the parameter space of all rational maps. In 1982, Thurston gave a topological characterization of PCF rational functions, and proved that they are "rigid", i.e. do not deform. This influential result was proven by realizing every PCF rational function as the unique fixed point of a holomorphic dynamical system on the Teichm?ller space of a punctured sphere. I will give an overview of the study of PCF maps, and discuss how, by a reformulation (due to Koch) of Thurston's maps on Teichm?ller space, every PCF map arises as an isolated fixed point of an algebraic dynamical system on the moduli space M_{0,n} of point-configurations on the Riemann sphere. I will then present results on the complexity of the ensuing dynamics on moduli space.
Location: Zoom

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