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The multicorns are the connectedness loci of unicritical antiholomorphic polynomials z¯d+c. We investigate the structure of boundaries of hyperbolic components: we prove that the structure of bifurcations from hyperbolic components of even period is as one would expect for maps that depend holomorphically on a complex parameter (for instance, as for the Mandelbrot set; in this setting, this is a non-obvious fact), while the bifurcation structure at hyperbolic components of odd period is very different. In particular, the boundaries of odd period hyperbolic components consist only of parabolic parameters, and there are bifurcations between hyperbolic components along entire arcs, but only of bifurcation ratio 2. We also count the number of hyperbolic components of any period of the multicorns. Since antiholomorphic polynomials depend only real-analytically on the parameters, most of the techniques used in this paper are quite different from the ones used to prove the corresponding results in a holomorphic setting.
| arXiv:1404.5031 |
It is well known that every rational parameter ray of the Mandelbrot set lands at a single parameter. We study the rational parameter rays of the multicorn ∗d, the connectedness locus of unicritical antiholomorphic polynomials of degree d, and give a complete description of their accumulation properties. One of the principal results is that the parameter rays accumulating on the boundaries of odd period (except period 1) hyperbolic components of the multicorns do not land, but accumulate on arcs of positive length consisting of parabolic parameters.
We also show the existence of undecorated real-analytic arcs on the boundaries of the multicorns, which implies that the centers of hyperbolic components do not accumulate on the entire boundary of ∗d, and the Misiurewicz parameters are not dense on the boundary of ∗d.
A physical measure on the attractor of a system describes the statistical behavior of typical orbits. An example occurs in unimodal dynamics. Namely, all infinitely renormalizable unimodal maps have a physical measure. For Lorenz dynamics, even in the simple case of infinitely renormalizable systems, the existence of physical measures is more delicate. In this article we construct examples of infinitely renormalizable Lorenz maps which do not have a physical measure. A priori bounds on the geometry play a crucial role in (unimodal) dynamics. There are infinitely renormalizable Lorenz maps which do not have a priori bounds. This phenomenon is related to the position of the critical point of the consecutive renormalizations. The crucial technical ingredient used to obtain these examples without a physical measure, is the control of the position of these critical points.
We prove that any two $C^4$ critical circle maps with the same irrational rotation number and the same odd criticality are conjugate to each other by a $C^1$ circle diffeomorphism. The conjugacy is $C^{1+\alpha}$ for Lebesgue almost every rotation number.
We prove that the parameter space of monic centered cubic polynomials with a critical point of exact period n = 4 is connected. The techniques developed for this proof work for every n and provide an interesting relation between escaping trees of DeMarco-McMullen and Hubbard trees.
In the family of area-contracting Henon-like maps with zero topological entropy we show that there are maps with infinitely many moduli of stability. Thus one cannot find all the possible topological types for non-chaotic area-contracting Hénon-like maps in a family with finitely many parameters. A similar result, but for the chaotic maps in the family, became part of the folklore a short time after Hénon used such maps to produce what was soon conjectured to be the first non-hyperbolic strange attractors in $\mathbb{R}^2$. Our proof uses recent results about infinitely renormalisable area-contracting Hénon-like maps; it suggests that the number of parameters needed to represent all possible topological types for area-contracting Hénon-like maps whose sets of periods of their periodic orbits are finite (and in particular are equal to $\{1,\, 2,\dots,\,2^{n-1}\}$ or an initial segment of this n-tuple) increases with the number of periods. In comparison, among $C^k$-embeddings of the 2-disk with $k>0$, the maximal moduli number for non-chaotic but non area-contracting maps in the interior of the set of zero-entropy is infinite.
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.
The error diffusion algorithm can be considered as a time dependent dynamical system that transforms a sequence of inputs; into a sequence of inputs;. That dynamical system is a time dependent translation acting on a partition of the phase space $\mathbb{A}$, a finite dimensional real affine space, into the Voronoï regions of the set $C$ of vertices of some polytope $\mathbf {P}$ where the inputs all belong.
Given a sequence $g(i)$ of inputs that are point in $\mathbb{A}$, $g(i)$ gets added to the error vector $e(i)$, the total vector accumulated so far, that belongs to the (Euclidean) vector space mofelling $\mathbb{A}$. The sum $g(i)+e(i)$ is then again in $\mathbb{A}$, thus in a well defined element of the partition of $\mathbb{A}$ that determines in turns one vertex $v(i)$. The point $v(i)$ of $\mathbb{A}$ is the $i^\textrm{th}$ output, and the new error vector to be used next is $e(i+1)\,=\, g(i)+e(i)-v(i)$. The maps $e(i)\mapsto e(i+1)$ and $g(i)+e(i)\mapsto g(i+1)+e(i+1)$ are two form of error diffusion, respectively in the vector space and affine space. Long term behavior of the algorithm can be deduced from the asymptotic properties of invariant sets, especially from the absorbing ones that serve as traps to all orbits. The existence of invariant sets for arbitrary sequence of inputs has been established in full generality, but in such a context, the invariant sets that are shown to exist are arbitrarily large and only few examples of minimal invariant sets can be described. Since the case of constant input (that corresponds to a time independent translation) has its own interest, we study here the invariant set for constant input for special polytopes that contain the $n$-dimensional regular simplices.
In that restricted context of interest in number theory, we study the properties of the minimal absorbing invariant set and prove that typically those sets are bounded fundamental sets for a discrete lattice generated by the simplex and that the intersections of those sets with the elements of the partition are fundamental sets for specific derived lattices.
We prove a version of the classical $\lambda$-lemma for holomorphic families of Riemann surfaces. We then use it to show that critical loci for complex Hénon maps that are small perturbations of quadratic polynomials with Cantor Julia sets are all quasiconformally equivalent.