JournalsjemsVol. 19, No. 11pp. 3421–3465

The dynamical Manin–Mumford problem for plane polynomial automorphisms

  • Romain Dujardin

    Université Paris-Est Marne-la-Vallée, Champs-sur-Marne, France
  • Charles Favre

    École Polytechnique, Palaiseau, France
The dynamical Manin–Mumford problem for plane polynomial automorphisms cover
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Abstract

Let ff be a polynomial automorphism of the affine plane. In this paper we consider the possibility for it to possess infinitely many periodic points on an algebraic curve CC. We conjecture that this happens if and only if ff admits a time-reversal symmetry; in particular the Jacobian Jac (f)(f) must be a root of unity.

As a step towards this conjecture, we prove that the Jacobian and all its Galois conjugates lie on the unit circle in the complex plane. Under mild additional assumptions we are able to conclude that indeed Jac (f)(f) is a root of unity.

We use these results to show in various cases that any two automorphisms sharing an infinite set of periodic points must have a common iterate, in the spirit of recent results by Baker–DeMarco and Yuan–Zhang.

Cite this article

Romain Dujardin, Charles Favre, The dynamical Manin–Mumford problem for plane polynomial automorphisms. J. Eur. Math. Soc. 19 (2017), no. 11, pp. 3421–3465

DOI 10.4171/JEMS/743