
## 2. Problems on Complex Dynamics

1. #### Problem 2.1.

Let $F$ be a non-polynomial 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 non-zero entropy. Adapt the study of critical measure, etc from the study of Hénon maps to that of $F$.

Cite this as: AimPL: Problems on holomorphic function spaces and complex dynamics, available at http://aimpl.org/scvproblems.