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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.
We extend the renormalization operator introduced in [3] from period-doubling Hénon-like maps to Hénon-like maps with arbitrary stationary combinatorics. We show the renormalisation prodcudure also holds in this case if the maps are taken to be strongly dissipative. We study infinitely renormalizable maps F and show they have an invariant Cantor set O on which F acts like a p-adic adding machine for some p > 1. We then show, as for the period-doubling case in [3], the sequence of renormalisations have a universal form, but the invariant Cantor set O is non-rigid. We also show O cannot possess a continuous invariant line field.
We show that given a one parameter family $F_b$ of strongly dissipative infinitely renormalisable Hénon-like maps, parametrised by a quantity called the 'average Jacobian' b, the set of all parameters b such that $F_b$ has a Cantor set with unbounded geometry has full Lebesgue measure.
We prove exponential contraction of renormalization along hybrid classes of infinitely renormalizable unimodel maps (with arbitrary combinatorics), in any even degree d. We then conclude that orbits of renormalization are asymptotic to the full renormalization horseshoe, which we construct. Our argument for exponential contraction is based on a precompactness property of the renormalization operator ("beau bounds"), which is leveraged in the abstract analysis of holomorphic iteration. Besides greater generality, it yields a unified approach to all combinatorics and degrees: there is no need to account for the varied geometric details of the dynamics, which were the typical source of contraction in previous restricted proofs.