Polynomial skew products with small relative degree

Abstract

We investigate the local dynamics of a proper superattracting holomorphic germ $f$ in $(\mathbb{C}^2,0)$ possessing a totally invariant line $L$ such that $f^*L = d L$ with $d\ge 2$, and such that $f|_L$ has a superattracting fixed point at $0$ of order $2 \le c < d$. We prove that any such map is formally conjugated to a skew product of the form $(z^d, P(z,w))$, where $P \in \mathbb{C}[[z]][w]$ is polynomial in $w$ of degree $c$, hence it induces a natural dynamics on the Berkovich affine line over $\mathbb{C}(\!(z)\!)$. Such non-Archimedean skew products were recently studied by Birkett and Nie-Zhao. On the non-Archimedean side, we focus on the restriction of the dynamics on the Berkovich open unit ball (which naturally contains all irreducible analytic germs at the origin). We exhibit an invariant compact set $\mathcal{K}$ outside of which all points tend to $L$, and which supports a natural ergodic invariant measure. By a careful analysis of local intersection numbers, we prove that the growth of multiplicity of iterated curves is controlled by the recurrence properties of the critical set. In particular, when no critical branch of $f$ belongs to $\mathcal{K}$, any point in $\mathcal{K}$ corresponds to a curve of uniformly bounded multiplicity at $0$. We then return to the complex picture and show the existence of an invariant pluripolar positive closed $(1,1)$-current $T$, outside of which all orbits converge to $0$ at super-exponential speed $c$. Under the same assumption on the critical branches as above, we prove that $T$ admits a geometric representation as an average of currents of integration over the curves in $\mathcal{K}$, with respect to the natural invariant measure. In particular, $T$ is uniformly laminar outside the origin.