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 $f:CtoC$ entire transcendental  and $ainR$.

• 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 $ ho:Omega ightarrowSigmasubset 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.