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2. Moduli Problems

    1. Equidistribution Problems


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      Problem 2.1.

      1. f:\mathbb{P}^1 \to \mathbb{P}^1. Is the set PCF equidistributed on M_d with respect to the bifurcation measure.
      2. Same question for f:\mathbb{P}^N \to \mathbb{P}^N with respect to what measure?
          Part (1) is known for polynomials - Thomas Gauthier and Charles Favre

        • Critical orbit relations


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          Conjecture 2.2.

          [Lauran DeMarco/Patrick Ingram] An algebraic subvariety in M_d has a Zariski dense subset of PCF maps if and only if that subvariety is defined by critical orbit relations.

          1. Let X \subset \mathbb{A}^2 be a curve. Is it true that you have infinitely many points (c_1,c_2) \in X such that z^2+c_1 and z^2+c_2 are both PCF if and only if it is vertical line, a horizontal line, or the diagonal?

            • PCF locus


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              Problem 2.3.

              1. How big is the PCF locus in M_d^N?
              2. Given a “critical portrait” when is the set of PCF maps with that portrait zero dimensional? (what are the obvious counter examples?)

                • p-adic PCF locus


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                  Problem 2.4.

                  Is M_d(\Q_p) \cap PCF (or polynomials P_d(\Q_p) \cap PCF) infinite for all p? (maybe no for p > d, but what about p < d?) What are the accumulation points?

                    • regular polynomial endomorphisms


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                      Problem 2.5.

                      [Patrick Ingram] Fix d,N. f:\mathbb{P}^N \to \mathbb{P}^N is a regular polynomial endomorphism if it leaves a hyperplane invariant, call the space of them R_d^N. If we restrict each f to its hyperplane it gives a projection \pi: R_d^N \to M_d^{N-1}. In any given fiber of \pi is there a curve of PCF maps? (Known: Ingram-There are horizontal families, i.e. intersect every fiber in finitely many places. There are no PCF maps above the power map).

                      Alternate formulation: If you have a map from a curve C \to R_d^N \to M_d^{N-1} where the map C \to M_d^{N-1} is constant and C lands in the PCF locus of R_d^N. Is the map C \to R_d^N already constant?

                        • field of definition vs field of moduli


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                          Problem 2.6.

                          Are all PCF maps defined over their field of moduli?

                            • Classification of moduli space


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                              Problem 2.7.

                              [Joseph Silverman] Fix d. If n >>_d 1, then is M_d(n) of general type, where M_d(n) marks periodic points of formal period n.
                                  Known for abelian varieties and for d=2, n=1,\ldots,5 M_d(n) is rational and M_2(6) is general type (Blanc-Canci)

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