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PREPRINTS IN THIS SERIES, IN PDF FORMAT.
* Starred papers have appeared in the journal cited.

A structure theorem for semi-parabolic Henon maps
Abstract:
Abstract:
Consider the parameter space $\mathcal{P}_{\lambda}\subset \mathbb{C}^{2}$ of complex Hénon maps $H_{c,a}(x,y)=(x^{2}+c+ay, ax),\ a\neq 0$ which have a semi-parabolic fixed point with one eigenvalue $\lambda=e^{2\pi i p/q}$. We give a characterization of those Hénon maps from the curve $\mathcal{P}_{\lambda}$ that are small perturbations of a quadratic polynomial $p$ with a parabolic fixed point of multiplier $\lambda$. We prove that there is an open disk of parameters in $\mathcal{P}_{\lambda}$ for which the semi-parabolic Hénon map has connected Julia set $J$ and is structurally stable on $J$ and $J^{+}$. The Julia set $J^{+}$ has a nice local description: inside a bidisk $\mathbb{D}_{r}\times \mathbb{D}_{r}$ it is a trivial fiber bundle over $J_{p}$, the Julia set of the polynomial $p$, with fibers biholomorphic to $\mathbb{D}_{r}$. The Julia set $J$ is homeomorphic to a quotiented solenoid.
R. Dujardin, M. Lyubich
Stability and bifurcations of dissipative polynomial automorphisms of $\mathbb{C}^2$
Abstract:

We study stability and bifurcations in holomorphic families of polynomial automorphisms of $\mathbb{C}^2$. We say that such a family is weakly stable over some parameter domain if periodic orbits do not bifurcate there. We first show that this defines a meaningful notion of stability, which parallels in many ways the classical notion of J-stability in one-dimensional dynamics. In the second part of the paper, we prove that under an assumption of moderate dissipativity, the parameters displaying homoclinic tangencies are dense in the bifurcation locus. This confirms one of Palis' Conjectures in the complex setting. The proof relies on the formalism of semi-parabolic bifurcation and the construction of "critical points" in semiparabolic basins (which makes use of the classical Denjoy-Carleman-Ahlfors and Wiman Theorems).

E. Bedford, J. Smillie and T. Ueda
Parabolic Bifurcations in Complex Dimension 2
Abstract:

In this paper we consider parabolic bifurcations of families of diffeomorphisms in two complex dimensions.

J. Milnor, with an appendix by A. Poirier
Hyperbolic Components
Abstract:

Consider polynomial maps $f : \mathbb{C} \to \mathbb{C}$ of degree $d \geq 2$, or more generally polynomial maps from a finite union of copies of $\mathbb{C}$ to itself. In the space of suitably normalized maps of this type, the hyperbolic maps form an open set called the hyperbolic locus. The various connected components of this hyperbolic locus are called hyperbolic components, and those hyperbolic components with compact closure (or equivalently those contained in the "connectedness locus") are called bounded hyperbolic components. It is shown that each bounded hyperbolic component is a topological cell containing a unique post-critically finite map called its center point. For each degree d, the bounded hyperbolic components can be separated into finitely many distinct types, each of which is characterized by a suitable reduced mapping scheme $S_f$. Any two components with the same reduced mapping scheme are canonically biholomorphic to each other. There are similar statements for real polynomial maps, for polynomial maps with marked critical points, and for rational maps. Appendix A, by Alfredo Poirier, proves that every reduced mapping scheme can be represented by some classical hyperbolic component, made up of polynomial maps of $\mathbb{C}$. This paper is a revised version of [M2], which was circulated but not published in 1992.

J. Milnor
Arithmetic of Unicritical Polynomial Maps
Abstract:

This note will study complex polynomial maps of degree $n \geq 2$ with only one critical point.

D. Gaidashev, T. Johnson, M. Martens
Rigidity for infinitely renormalizable area-preserving maps
Abstract:

Area-preserving maps have been observed to undergo a universal period-doubling cascade, analogous to the famous Feigenbaum-Coullet-Tresser period doubling cascade in one-dimensional dynamics. A renormalization approach has been used by Eckmann, Koch and Wittwer in a computer-assisted proof of existence of a conservative renormalization fixed point. Furthermore, it has been shown that infinitely renormalizable maps in a neighborhood of this fixed point admit invariant Cantor sets on which the dynamics is "stable" - the Lyapunov exponents vanish on these sets.

Infinite renormalizability exists in several settings in dynamics, for example, in unimodal maps, dissipative Hénon-like maps, and conservative Hénon-like maps. All of these types of maps have associated invariant Cantor sets. The unimodal Cantor sets are rigid: the restrictions of the dynamics to the Cantor sets for any two maps are $C^{1+\alpha}$-conjugate. Although, strongly dissipative Hénon maps can be seen as perturbations of unimodal maps, surprisingly the rigidity breaks down. The Cantor attractors of Hénon maps with different average Jacobians are not smoothly conjugated. It is conjectured that the average Jacobian determines the rigidity class. This conjecture holds when the Jacobian is identically zero, and in this paper we prove that the conjecture also holds for conservative maps close to the conservative renormalization fixed point.

Rigidity is a consequence of an interplay between the decay of geometry and the convergence rate of renormalization towards the fixed point. Therefore, to demonstrate rigidity, we prove that the upper bound on the spectral radius of the action of the renormalization derivative on infinitely renormalizable maps is sufficiently small.

M. Martens and B. Winckler
On the Hyperbolicity of Lorenz Renormalization
Abstract:

We consider infinitely renormalizable Lorenz maps with real critical exponent $\alpha>1$ and combinatorial type which is monotone and satisfies a long return condition. For these combinatorial types we prove the existence of periodic points of the renormalization operator, and that each map in the limit set of renormalization has an associated unstable manifold. An unstable manifold defines a family of Lorenz maps and we prove that each infinitely renormalizable combinatorial type (satisfying the above conditions) has a unique representative within such a family. We also prove that each infinitely renormalizable map has no wandering intervals and that the closure of the forward orbits of its critical values is a Cantor attractor of measure zero.

A. M. Benini, M. Lyubich
Repelling periodic points and landing of rays for post-singularly bounded exponential maps
Abstract:

We show that repelling periodic points are landing points of periodic rays for exponential maps whose singular value has bounded orbit. For polynomials with connected Julia sets, this is a celebrated theorem by Douady, for which we present a new proof. In both cases we also show that points in hyperbolic sets are accessible by at least one and at most finitely many rays. For exponentials this allows us to conclude that the singular value itself is accessible.

M. Lyubich and H. Peters
Classification of invariant Fatou components for dissipative Henon maps
Abstract:

Fatou components for rational functions in the Riemann sphere are very well understood and play an important role in our understanding of one-dimensional dynamics. In higher dimensions the situation is less well understood. In this work we give a classification of invariant Fatou components for moderately dissipative Hénon maps. Most of our methods apply in a much more general setting. In particular we obtain a partial classification of invariant Fatou components for holomorphic endomorphisms of projective space, and we generalize Fatou's Snail Lemma to higher dimensions.

F. Berteloot, T. Gauthier
On the geometry of bifurcation currents for quadratic rational maps
Abstract:

We describe the behaviour at infinity of the bifurcation current in the moduli space of quadratic rational maps. To this purpose, we extend it to some closed, positive (1,1)-current on a two-dimensional complex projective space and then compute the Lelong numbers and the self-intersection of the extended current.