A dichotomy for Fatou components of polynomial skew products

Abstract

Abstract. We consider polynomial maps of the form f (z, w) = (p(z), q(z, w)) that extend as holomorphic maps of CP 2 . Mattias Jonsson introduces in "Dynamics of polynomial skew products on C 2 " [Math. Ann., 314(3): 403-447, 1999] a notion of connectedness for such polynomial skew products that is analogous to connectivity for the Julia set of a polynomial map in one-variable. We prove the following dichotomy: if f is an Axiom-A polynomial skew product, and f is connected, then every Fatou component of f is homeomorphic to an open ball; otherwise, some Fatou component of F has infinitely generated first homology.

An implosion arising from saddle connection in 2D complex dynamics

“…The first statement follows from the stable manifold theorem for the hyperbolic set J 0 . See the proof of Theorem 1.2 in [13]. The local stable manifold W s loc (J 0 ) of J 0 is the union of local stable curves W s loc (w), w ∈ J 0 .…”

“…The basic properties of the holomorphic dynamics of polynomial skew products were established by Jonsson [7] and developed by many authors since then (see DeMarco-Hruska [4] and Roeder [13] for example). On the other hand, in the theory of the dynamics of Axiom A maps, the relations between basic sets play a central role, which is defined through the intersection of the stable set of a basic set and the unstable set of another one.…”

Polynomial skew-products with wandering Fatou-disks

“…The topology of Fatou components of skew products has been studied by Roeder in [14]. Since skew-products map vertical lines to vertical lines, often one-dimensional tools can be used to construct maps with specific dynamical behavior.…”

“…The holomorphic dynamics of polynomial skew-products was first studied by Heinemann in [8] and by Jonsson in [7]. The topology of Fatou components of skew products has been studied by Roeder in [12]. Since skew-products map vertical lines to vertical lines, often one-dimensional tools can be used to construct maps with specific dynamical behavior.…”

Polynomial skew-products in dimension 2: Bulging and Wandering Fatou components

“…The investigation of the holomorphic dynamics of polynomial skew-products was started by Heinemann [16] and then continued by Jonsson [18]. The topology of Fatou components of skew-products has been studied by Roeder in [26].…”