Welcome to the preprint server of the Institute for Mathematical Sciences at Stony Brook University.
The IMS preprints are also available from the mathematics section of the arXiv e-print server, which offers them in several additional formats. To find the IMS preprints at the arXiv, search for Stony Brook IMS in the report name.
PREPRINTS IN THIS SERIES, IN PDF FORMAT.
* Starred papers have appeared in the journal cited.
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.
This note will study complex polynomial maps of degree $n \geq 2$ with only one critical point.
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.
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.
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.
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.
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.
We give a topological model of the critical locus for complex Hénon maps that are perturbations of the quadratic polynomial with disconnected Julia set.
In this paper we continue to explore infinitely renormalizable Hénon maps with small Jacobian. It was shown in [CLM] that contrary to the one-dimensional intuition, the Cantor attractor of such a map is non-rigid and the conjugacy with the one-dimensional Cantor attractor is at most 1/2-Hölder. Another formulation of this phenomenon is that the scaling structure of the Hénon Cantor attractor differs from its one-dimensional counterpart. However, in this paper we prove that the weight assigned by the canonical invariant measure to these bad spots tends to zero on microscopic scales. This phenomenon is called Probabilistic Universality. It implies, in particular, that the Hausdorff dimension of the canonical measure is universal. In this way, universality and rigidity phenomena of one-dimensional dynamics assume a probabilistic nature in the two-dimensional world.
In a classical work of the 1950's, Lee and Yang proved that for fixed nonnegative temperature, the zeros of the partition functions of a ferromagnetic Ising model always lie on the unit circle in the complex magnetic field. Zeros of the partition function in complex temperature were then considered by Fisher, when the magnetic field is set to zero. Limiting distributions of Lee-Yang and of Fisher zeros are physically important as they control phase transitions in the model. One can also consider the zeros of the partition function simultaneously in both complex magnetic field and complex temperature. They form an algebraic curve called the Lee-Yang-Fisher (LYF) zeros. In this paper we study their limiting distribution for the Diamond Hierarchical Lattice (DHL). In this case, it can be described in terms of the dynamics of an explicit rational function R in two variables (the Migdal-Kadanoff renormalization transformation). We prove that the Lee-Yang-Fisher zeros are equidistributed with respect to a dynamical (1,1)-current in the projective space. The free energy of the lattice gets interpreted as the pluripotential of this current. We also describe some of the properties of the Fatou and Julia sets of the renormalization transformation.