摘  要：Given a marked point of an algebraic family of rational maps on the projective line, there are infinitely many preperiodic points by Montel's theorem. As a special case, there are infinitely many torsion points on a section of a family of elliptic curves. Now given a family of simple abelian varieties of relative dimension larger than 1, the subset of torsion points are not Zariski dense in any non-torsion section, by a recent result of Gao and Habegger. We propose a dynamical analogue of this result: given a family of plane regular polynomial automorphisms of the affine plane parameterized by a curve over a number field, we show that there are only finitely many parameters for which the marked point is periodic, if the Jacobian of the family is constant of modulus different than 1.