PSU Mathematics Department - Meeting Details

For more information about this meeting, contact Anatole Katok, Dmitri Burago.

Abstract:
Title:	Algebraic independence of multipliers of two periodic orbits in the space of polynomials of degree greater or equal than three
Seminar:	Center for Dynamics and Geometry Seminar
Speaker:	Igor Gorbovickis, Cornell University
Let $P_d$ be the space of monic polynomials of degree $d\ge 3$. For any pair of distinct periodic orbits of a polynomial from $P_d$, consider a corresponding pair of (multi valued) algebraic functions on $P_d$, obtained by analytic continuation of the multipliers of these orbits. In the first part of the talk we will show that these two functions are algebraically independent. A map is Kupka-Smale if all its periodic points are hyperbolic and the stable and unstable manifolds of any two saddle points are transverse. In the second part of the talk we will show, how the result from the first part is used in the proof of the Kupka-Smale theorem for volume preserving polynomial automorphisms of $\bbC^2$ of dynamical degree $d\ge 3$. Roughly speaking, the theorem says that a typical volume preserving polynomial automorphisms of $\bbC^2$ of dynamical degree $d\ge 3$ is Kupka-Smale.