Motivated by our recent works around the Dynamical Manin-Mumford problem for polynomial endomorphisms of C^2, we investigate the local dynamics of polynomial skew products of the form (z^d, w^c + zh(z,w)), under the condition 2 <= c < d (small relative degree).

For these maps, the asymptotic contraction rate of any point p exists and is either c or d.

We show that the locus W where the latter situation happens, analogous of the super-stable manifold in our setting, is the support of a pluripolar the Green current T introduced by Favre-Jonsson.

Moreover, under a natural condition on the dynamics of the critical branches, we describe T as an average of currents of integration along a Cantor set of holomorphic curves.

This structure can be elegantly interpreted through the induced dynamics on the Berkovich affine line over the field of Laurent series.

This is a joint work with Romain Dujardin and Charles Favre.