Wandering Fatou components near invariant fibers beyond the treated elliptic case

Determine whether wandering Fatou components can occur near invariant fibers of polynomial skew products in the general invariant-fiber setting, beyond the parabolic examples and the linearizable elliptic setting treated here.

Background

The paper considers Raissy’s question concerning the possible occurrence of wandering Fatou components near invariant fibers of polynomial skew products. It notes that the question encompasses all types of invariant fibers and that positive examples are already known in the parabolic case.

The paper resolves the question only in a restricted linearizable elliptic setting: it proves a local no-wandering theorem for elliptic polynomial skew products over bounded rotation domains carrying an absolutely continuous invariant probability measure of full support, and derives a global criterion for regular skew products whose bounded base Fatou components eventually enter periodic Siegel disks. The broader invariant-fiber question therefore remains unresolved outside these treated regimes, including nonlinearizable elliptic bases.