In this mini-course, I will present some of the recent results obtained in collaboration with Romain Dujardin and Charles Favre.

Motivated by recent works around the Dynamical Manin-Mumford problem for polynomial endomorphisms of the complex plane, we investigate the local dynamics of polynomial skew products of the form ($z^d$, $P(z,w)$), where $P ∈ C[[z]][w]$ is a monic polynomial in w of degree 2 <= c < d (small relative degree).

In particular, we show that the asymptotic contraction rate of any point p in a neighborhood of the origin exists, and is equal to either c or d.

The locus W where the latter situation happens, analogous of the super-stable manifold in our setting, is the support of a pluripolar positive closed (1,1)-current T.

In order to study its properties, we analyze the induced dynamics f_⋄ on the Berkovich affine line over C((z)) (see also the recent works by Birkett and Nie-Zhao).

In the non-archimedean setting, W is strictly related to the Julia set K of f_⋄, which is the locus of points in the unit ball not converging towards the Gauss point ζ_g.

By a careful analysis of local intersection numbers, we prove that the growth of multiplicity along orbits in K is controlled by the recurrence properties of the (twisted) critical set.

In particular, when no critical branches belong to K, then K corresponds to curves of uniformly bounded multiplicity.

In this case, we show that T admits a geometric representation as an average of currents of integration over the curves in K with respect to a natural invariant measure.

The course will take place from 12th to 14th November, with daily sessions commencing at 2:00 PM and concluding at 4:00 PM.