Complex Analysis, Dynamics and Geometry Seminar

It is well-known the parabolic polynomials on the boundary of the main cardioid of the Mandelbrot set are in one-to-one correspondence with a rational rotation number; two such parabolic polynomials are mateable if and only if they have non-conjugate rotation number; and the boundary is a Jordan curve.
In this talk, we will study these three statements for the principal hyperbolic component (the hyperbolic component of polynomials containing z^d) of higher degree. We will classify the geometrically finite polynomials on the boundary of the main hyperbolic component, and characterize their mateablity condition. We also find a new self-bump phenomenon on the boundary of principal hyperbolic component of higher degree: a geometrically finite polynomial that is accessed in two different ways from the principal hyperbolic component. Speaker(s): Yusheng Luo (U(M))