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7. Dynatomic side

Let $K$ be a number field $f(z)\in K(z)$ $\deg f\geq 2.$ Let \[ K_f=K(\text{all preperiodic points of }f). \]
    1. Problem 7.1.

      [Nguyen] If $K_fK_g$ is finite over both $K_f$ and $K_g$, are $f$ and $g$ conjugate, an iterate, or conjugate to something that commutes?
        • Problem 7.2.

          What do you need to know about $K_f$ to figure out what $f$ was (or how close to $f$ can you get)?
            • Problem 7.3.

              [Silverman] Begin to classify those $f$ for which $\text{Gal}(K_f/K)$ does not have finite index in the thing it’s supposed to. Ask this question again for \[ K_f'=K(\text{all periodic points of }f) \]

                  Cite this as: AimPL: The Galois theory of orbits in arithmetic dynamics, available at http://aimpl.org/galarithdyn.