Dynamical Systems seminar - Degenerations and irreducibility problems in rational dynamics

Dates

Wednesday, May 3, 2023 - 14:00 to 15:00

Speaker: Rohini Ramadas (University of Warwick)

Abstract: Per_n is a (nodal) Riemann surface parametrizing degree-2 rational functions with an n-periodic critical point. The n-th Gleason polynomial G_n is a polynomial in one variable with Z-coefficients, whose roots correspond to degree-2 polynomials with an n-periodic critical point (i.e. to the period-n components of the Mandelbrot set). Two long-standing open questions are: (1) Is Per_n connected? (2) Is G_n is irreducible over Q? We show that if G_n is irreducible over Q, then Per_n is connected. In order to do this, we find a smooth point with Q-coordinates on a compactification of Per_n. This smooth Q-point represents a special degeneration of degree-2 rational maps, and as such admits an interpretation in terms of tropical geometry.