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5. Higher dimensions

Problems related to the notion of PCF for endomorphism of $\mathbb{P}^N$ for $N>1$.
    1. Notions of PCF

          $f:\mathbb{P}^N \to \mathbb{P}^N$ morphism.
      1. Defn: PCF if the orbit of the ramification locus $C$, $V=\bigcup_{n \geq 0} f^n(C)$ is a proper algebraic subvariety of $\mathbb{P}^N$.
      2. Defn: “PCF all the way down”: If the restriction of $f$ to every periodic component of the post-critical component $V$ is PCF.

      Problem 5.1.

      1. Does (1) imply (2)?
      2. Are PCF (1) or (2) Zariski dense in $M_d^N$?
      3. Is there are more general definition than (it1) such that:

        a) “there are more of them”? (i.e. less restrictive)

        b) they are dense in any reasonable topology

        c) Andre-Oort property

        • Thurston Rigidity Exceptions

          Problem 5.2.

          What are the “obvious” exception to Thurston Rigidity in $M_d^N$?
            • Flexible Family of PCF maps

              Problem 5.3.

              Is there a flexible family of PCF maps that does not come from $\mathbb{P}^1$.
                  resolved: yes.

                  Cite this as: AimPL: Postcritically finite maps in complex and arithmetic dynamics, available at http://aimpl.org/finitedynamics.