Algebraic Dynamics seminar: Critical Locus and Complex Solenoids for Polynomial Automorphisms of C^2

For one-dimensional holomorphic maps, orbits of the critical points largely determine the dynamics. Polynomial automorphisms of C^2, being invertible, do not possess critical points in the classical sense. However, using the fact that critical points of the polynomials are also critical points of the associated Green’s function, Hubbard, Bedford and Smillie defined critical loci, a meaningful 2-dimensional analog of the critical points. Bedford and Smillie used critical loci to define critical measures and prove an analog of Manning-Przytycki formula. We will examine critical loci in dynamically significant regions and explain how they relate to the connectivity properties and structure of Julia sets. We’ll also discuss Hubbard’s conjecture concerning the critical locus for maps with connected Julia sets and its relation to complex solenoids.