The dynamical Manin-Mumford problem for plane polynomial automorphisms (Q1676090)

Given a polynomial automorphism \(f\) of the affine plane, the authors seek to identify conditions under which \(f\) possesses infinitely many periodic points on some algebraic curve \(C\). They conjecture that this happens if and only if some iterate of \(f\) is reversible, i.e., it is conjugate to its inverse via an involution. Under such a circumstance, the Jacobian of \(f\) must be a root of unity. This paper makes notable progress towards this conjecture. The authors show that for automorphisms of Hénon type (the only non-trivial case) defined over a field of characteristics zero, the existence of an algebraic curve \(C\) containing infinitely many periodic points of \(f\) implies that the Jacobian of \(f\) and all its Galois conjugates have unit complex modulus. They also prove that if the regular locus of \(C\) contains a periodic point \(p\) such that the tangent to \(C\) at \(p\) is not periodic under the induced action of \(f\), then the Jacobian of \(f\) is indeed a root of unity. Under the more restrictive assumption that \(p\) is a saddle point at an Archimedean place, the same result is established with a purely Archimedean argument. An application of these results leads to the characterisation of pairs \(f,g\) of automorphisms of Hénon type which share an infinite set of periodic points. The authors prove that \(f\) and \(g\) must have a common iterate in the following two cases: i) \(f\) and \(g\) are defined over a number field, and their common periodic points are Zariski dense, and ii) \(f\) and \(g\) have complex coefficients and the modulus of the Jacobian of \(f\) is not equal to 1. These results echo recent work by \textit{M. Baker} and \textit{L. DeMarco} [Duke Math. J. 159, No. 1, 1--29 (2011; Zbl 1242.37062)] and \textit{X. Yuan} and \textit{S.-W. Zhang} [Math. Ann. 367, No. 3--4, 1123--1171 (2017; Zbl 1372.14017)].