Notions of Hyperbolicity in Complex Dynamics
Talk by Dr. Lukas Geyer (Mathematical Sciences, MSU)
4/15/2019 Wilson Hall 1-144 4:10-5:00pm
Complex dynamics studies the iteration of rational maps and polynomials in the complex plane. For any such map one can partition the plane into two invariant sets, the "tame" Fatou set and the "chaotic" Julia set. A map is said to be hyperbolic if it is expanding on its Julia set. Hyperbolicity is a desirable property because hyperbolic dynamics are very well-understood and stable under perturbations. I will explain the connection between hyperbolicity, forward orbits of critical points, and the hyperbolic metric of planar domains. In the second part of the talk I will introduce the weaker notions of subhyperbolicity and semihyperbolicity, which are defined in terms of forward orbits of critical points, as well as their connections to hyperbolic orbifold metrics, and how this leads to a slightly weaker form of expansion on their Julia sets.