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6. Other Questions

    1. Entropy


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      Problem 6.1.

      1. Let $f(z) = z^2 + c$, $c \in \R$ and PCF with invariant $[a,b]$ with fixed point at one end. Which algebraic numbers arise as $e^s$ where $s$ is the entropy of $f$?
      2. More generally, for $z^2+c$ on its Hubbard tree (or higher degree).
      3. Is there an algebraic analogue of entropy (like dynamical degree vs. arithmetic degree in higher dimensions)?

        • Milnor’s Characterisation of Lattes

              Milnor’s characterization of Lattès with 4 post-critical points
          • no critical point is post-critical
          • all critical points are simple

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          Problem 6.2.

          1. Does this hold over positive characteristic?
          2. What do you get from the dual of Frobenius when inseparable?
          3. What about characteristic 2 and 3 and $\mathop{Aut}(E)$ nonabelian?

            • Are there any more conspiracies?


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              Problem 6.3.

              [Adam Epstein] Let $\mathop{Per}_n(\lambda)$ be the locus in the moduli space of formal $n$-cycles with multiplier $\lambda$. $\mathop{Per}_n(\lambda)$ is in general an irreducible cubic, $\\mathop{Per}_3(1)$ factors. Let $\mathop{Per}^{\ast}_n(\lambda)$ be the same but with actual $n$-cycles instead of formal $n$-cycles. $\mathop{Per}^{\ast}_3(1) = \mathop{Per}_2(-3)$ are the same line in $\mathbb{A}^2$ (Milnor). $\mathop{Per}_n(\lambda) \cap \mathop{Per}_m(\lambda')$ should be zero dimensional (need at least one multiplier outside the unit circle).

              Are there any other examples $n,m,\lambda,\lambda'$ where the intersection is not zero dimensional?

                • portraits of rigid Lattès


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                  Problem 6.4.

                  What portraits occur for the rigid Lattès maps? In particular, are any of them the flexible Lattès portraits?

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