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In this paper we consider parabolic bifurcations of families of diffeomorphisms in two complex dimensions.
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.
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.
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 a classical work of the 1950's, Lee and Yang proved that the zeros of the partition functions of a ferromagnetic Ising model always lie on the unit circle. Distribution of the zeros is physically important as it controls phase transitions in the model. We study this distribution for the Migdal-Kadanoff Diamond Hierarchical Lattice (DHL). In this case, it can be described in terms of the dynamics of an explicit rational function $\mathcal{R}$ in two variables (the renormalization transformation). We prove that $\mathcal{R}$ is partially hyperbolic on an invariant cylinder $\mathcal{C}$. The Lee-Yang zeros are organized in a transverse measure for the central-stable foliation of $\mathcal{R}|\mathcal{C}$. Their distribution is absolutely continuous. Its density is $C^\infty$ (and non-vanishing) below the critical temperature. Above the critical temperature, it is $C^\infty$ on an open dense subset, but it vanishes on the complementary Cantor set of positive measure. This seems to be the first occasion of a complete rigorous description of the Lee-Yang distributions beyond 1D models.
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.
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 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.
The parameter space $S_p$ for monic centered cubic polynomial maps with a marked critical point of period p is a smooth affine algebraic curve whose genus increases rapidly with p. Each $S_p$ consists of a compact connectedness locus together with finitely many escape regions, each of which is biholomorphic to a punctured disk and is characterized by an essentially unique Puiseux series. This note with describe the topology of $S_p$, and of its smooth compactification, in terms of these escape regions. It concludes with a discussion of the real sub-locus of $S_p$.
We consider the family of holomorphic maps $e^z+c$ and show that fibers of postcritically finite parameters are trivial. This can be considered as the first and simplest class of non-escaping parameters for which we can obtain triviality of fibers in the exponential family.