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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.

      Add remark

      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


          Add remark

          Problem 5.2.

          What are the “obvious” exception to Thurston Rigidity in M_d^N?

            • Flexible Family of PCF maps


              Add remark

              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.