Loading Web-Font TeX/Math/Italic
| Register
\newcommand{\Cat}{{\rm Cat}} \newcommand{\A}{\mathcal A} \newcommand{\freestar}{ \framebox[7pt]{$\star$} }

6. Other Questions

    1. Entropy


      Add remark

      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

          Add remark

          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?


              Add remark

              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


                  Add remark

                  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.