Orbits Inside Basins of Attraction of Skew Products

A basic problem in complex dynamics is to understand orbits of holomorphic maps. One problem is to understand the collection of points $S$ in an attracting basin whose forward orbits land exactly on the attracting fixed point. In the paper [13], the second author showed that for holomorphic polynomials in $\mathbb C$, there is a constant $C$ so that all Kobayashi discs of radius $C$ must intersect this set $S$. In the paper [15], the second author showed that there are holomorphic skew products in $\mathbb {C}^2$ where this result fails. The main result of this paper is to show that for a large class of polynomial skew products, this result nevertheless holds.