Axiom A polynomial skew products of 2 and their postcritical sets

Abstract

A polynomial skew product of ℂ2 is a map of the form f(z,w)=(p(z),q(z,w)), where p and q are polynomials, such that f extends holomorphically to an endomorphism of ℙ2 of degree at least two. For polynomial maps of ℂ, hyperbolicity is equivalent to the condition that the closure of the postcritical set is disjoint from the Julia set; further, critical points either iterate to an attracting cycle or infinity. For polynomial skew products, Jonsson [Dynamics of polynomial skew products on C2. Math. Ann. 314(3) (1999), 403–447] established that f is Axiom A if and only if the closure of the postcritical set is disjoint from the right analog of the Julia set. Here we present an analogous conclusion: critical orbits either escape to infinity or accumulate on an attracting set. In addition, we construct new examples of Axiom A maps demonstrating various postcritical behaviors.

A dichotomy for Fatou components of polynomial skew products

“…In that case, the Implicit Function Theorem can be used to lift the entire holomorphic motion of J z k under f to a holomorphic motion of J z k+1 parameterized by U k+1 . The result will automatically satisfy (1) and (2).…”

“…We conclude this note by interpreting some of the examples of Axiom-A polynomial skew products from [12], [2], and [21] in the context of Theorem 1.2. is Axiom-A on CP 2 and is not in the same hyperbolic component as any product mapping. There are two reasons why f is disconnected:…”

“…We consider the dynamics of mappings f : C 2 → C 2 of the form f (z, w) = (p(z), q(z, w)) (1) maximal entropy µ (see §2, below) is connected, the Hausdorff dimension of µ is 2, and the associated Lyapunov exponents are precisely log d. (Note that J 2 can be connected even if f is a disconnected polynomial skew product; see [2,Lemma 5.5]. )…”

“…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).…”

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

“…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.…”