Some remarks on the connectivity of Julia sets for 2-dimensional diffeomorphisms

Abstract

Abstract. We explore the connected/disconnected dichotomy for the Julia set of polynomial automorphisms of C 2 . We develop several aspects of the question, which was first studied by Bedford-Smillie [BS6, BS7]. We introduce a new sufficient condition for the connectivity of the Julia set, that carries over for certain Hénon-like and birational maps. We study the structure of disconnected Julia sets and the associated invariant currents. This provides a simple approach to some results of Bedford-Smillie, as well as some new corollaries -the connectedness locus is closed, construction of external rays in the general case, etc.We also prove the following theorem: a hyperbolic polynomial diffeomorphism of C 2 with connected Julia set must have attracting or repelling orbits. This is an analogue of a well known result in one dimensional dynamics.

Stability and bifurcations for dissipative polynomial automorphisms of $${{\mathbb {C}}^2}$$ C 2

“…It is the counterpart of Theorem 8.3 in the dissipative 2-dimensional setting. Assuming that a semi-parabolic bifurcation of the form (19) occurs, it select parameters λ n such that the iterates f n λ n map a given point p ι in some semiparabolic basin (almost) onto a given target p o located in a repelling petal , with a good control on the geometry of f n λ n near z ι . This geometric control is expressed in terms of the pull-back action on a foliation transverse to near p o .…”

“…Normalize the situation so that f λ is locally of the form (19). Conjugating by a rotation, we may assume that the critical point lies in the basin B corresponding to the attracting direction {(x, 0), arg(x) = − π k }.…”

Continuity of Lyapunov exponents for polynomial automorphisms of $\mathbb{C}^2$

“…This condition is actually independent of the saddle point p [BS6] (see also [Du3,Prop. 1.8] Du3,Prop.…”

“…Laminar structure. In this paragraph we explain some results of [Du3] on the laminar structure of T − for unstably disconnected mappings. Let f be a polynomial automorphism satisfying the conclusion of Lemma 2.4.…”

“…1.8] Du3,Prop. 2.3] we proved that V is subordinate to T − in the sense that there exist a non trivial uniformly laminar current S ≤ T − , made up of a family of disjoint horizontal disks of degree deg(V ), extending V .…”