  We establish the finiteness of periodic points, that we called Geometric
Dynamical Northcott Property, for regular polynomials automorphisms of the
affine plane over a function field $\mathbf{K}$ of characteristic zero,
improving results of Ingram.
  For that, we show that when $\mathbf{K}$ is the field of rational functions
of a smooth complex projective curve, the canonical height of a subvariety is
the mass of an appropriate bifurcation current and that a marked point is
stable if and only if its canonical height is zero. We then establish the
Geometric Dynamical Northcott Property using a similarity argument.
