Dynamics of transcendental Henon maps [ Back ]

Dates:
19.09.17 - 22.11.17
Times:
09:00 to 11:00
Place:
Barcelona, Spain

 

Abstract:

In this course we cover the dynamics of transcendental Henon maps. A transcendental Henon map is an automorphisms of $C^2$ of the form $F(z,w)=(f(z)+aw,z)$ with $f:\C\to\C$ entire transcendental  and $a\in\R$.

  • Lecture 1. September 19. 9h-11h.
    Different definitions of normality for holomorphic functions in $\C^2$. Basic features of holomorphic  dynamics in $\C^2$. Existence of saddle points for transcendental Henon maps. Proof of the non-emptyness of the Julia set.
  • Lecture 2. September 20. 9h-11h.
    Classification of recurrent Fatou components. We show that for any invariant recurrent Fatou component $\Omega$ there is a retraction $\rho:\Omega \rightarrow\Sigma\subset \Omega$ where  $\Sigma$ is an invariant limit manifold of rank 0, 1 or 2. If $\Sigma $ has rank 0, $\Omega$ is an attracting domain; If $\Sigma$ has rank 2, $\Omega$ is a rotation domain; if $\Sigma$ has rank 1, then it is a rotational surface. 
  • Lecture 3. October 11. 9h-11h.
    Baker domains and subharmonic trick. We construct a transcendental Henon map $F$ with an invariant Baker domain on which $F$ is conjugate to a translation. This example is highly inspired by the construction of Baker domains in one-dimensional holomorphic dynamics. 
  • Lecture 4. November 22. 9h-11h.
    Escaping and Oscillating Wandering domains. We construct an example of a transcendental Henon map with a wandering domain whose orbits converge to infinity, and of a transcendental Henon map with n oscillating orbit of wandering domains.The first example is inspired by the construction of wandering domains in one variable while the second example is costructed using Runge approximation.