Topology of Fatou components for endomorphisms of CP^k: linking with the Green's current

Abstract

Little is known about the global topology of the Fatou set U (f) for holomorphic endomorphisms f : CP k → CP k , when k > 1. Classical theory describes U (f) as the complement in CP k of the support of a dynamically defined closed positive (1, 1) current. Given any closed positive (1, 1) current S on CP k , we give a definition of linking number between closed loops in CP k \ supp S and the current S. It has the property that if lk(γ, S) = 0, then γ represents a non-trivial homology element in H1(CP k \ supp S). As an application, we use these linking numbers to establish that many classes of endomorphisms of CP 2 have Fatou components with infinitely generated first homology. For example, we prove that the Fatou set has infinitely generated first homology for any polynomial endomorphism of CP 2 for which the restriction to the line at infinity is hyperbolic and has disconnected Julia set. In addition we show that a polynomial skew product of CP 2 has Fatou set with infinitely generated first homology if some vertical Julia set is disconnected. We then conclude with a section of concrete examples and questions for further study.

Towards a semi-local study of parabolic invariant curves for fibered holomorphic maps

“…Let {ε n } n∈N be a real sequence verifying ε n < (2n) ν (10) for some ν > 0. Let us define the sequence ϑ 1 , ϑ 2 , .…”

“…In recent years, special interest has been given to skew-product dynamical systems, and many results about complex skew-product dynamics have appeared (see [4,10,11,17,19,20]). On the one hand, skew-product dynamics presents a rich source of examples, and on the other hand, it is considered as an intermediate step towards higher-dimensional dynamics.…”

Superstable manifolds of invariant circles and codimension-one Böttcher functions

“…Let Ω n := {x : R −n {x} ⊆ Ω 0 } and C n be the critical value set for R n . For x ∈ Ω n \ C n , we may define (16) ϕ…”

“…This is described more precisely at the beginning of §2. Although this is a rather special situation, it has appeared in examples from [31,16,6,5]. For such maps, the Julia set of f |L is an invariant circle S, which is a hyperbolic set for f .…”

A dichotomy for Fatou components of polynomial skew products

“…For this reason, polynomial skew products provide an accessible generalization of one variable dynamics to two variables. They been previously studied by many authors, including Heinemann in [6, 7], Jonsson in [12], DeMarco-Hruska in [2], and Hruska together with the author of this note in [8].A more general situation in which the base is allowed to be an arbitrary compact topological space, while the vertical fibers are copies of C, has been considered by Sester in [17,18]. Meanwhile, generalization to semigroups of polynomial (and rational) mappings of the Riemann sphere has been studied by extensively by Hinkkanen, Martin, Ren, Stankewitz, Sumi, Urbański, and many others-we refer the reader to the excellent bibliography from [21] for further references.As an analogy with polynomial maps of one variable,Here, J p is the Julia set for the "base map" z → p(z).…”