2. Problems on Complex Dynamics

Problem 2.1.
Let $F$ be a nonpolynomial automorphism of $\mathbb{C}^2$. What are the possible biholomorphic types of Fatou components of $F$? 
Problem 2.2.
Investigate the dynamics of maps of the form \begin{equation*}(x, y) \mapsto (p(x)\delta y, x), \end{equation*} where $p(x)$ is a degree $2$ rational map. 
Problem 2.3.
Let $F$ be a germ of automorphism of $\mathbb{C}^n (n > 1)$ tangent to the identity with a degenerate characteristic direction $v$. When is there a domain of attraction tangent to the origin along $v$? 
Problem 2.4.
Study hedgehogs for gems of automorphism of $\mathbb{C}^2$ with both elliptic eigenvalues. 
Problem 2.5.
Let $C$ be the cubic surface defined by \begin{equation*} x^2 + y^2 + z^2  xyz = D, \end{equation*} where $D$ is a complex parameter. Let $s_x: C \rightarrow C$ be defined by \begin{equation*} s_x(x, y, z) := (yz  x, y, z). \end{equation*} The maps $s_y, s_z$ are defined similarly.
Consider, for example, the map $S:= s_x \circ s_y \circ s_z$. It is known that $S$ has nonzero entropy. Adapt the study of critical measure, etc from the study of HeĢnon maps to that of $F$.
Cite this as: AimPL: Problems on holomorphic function spaces and complex dynamics, available at http://aimpl.org/scvproblems.