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2. Problems on Complex Dynamics


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      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?


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          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.


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              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?


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                  Problem 2.4.

                  Study hedgehogs for gems of automorphism of \mathbb{C}^2 with both elliptic eigenvalues.


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                      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.