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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 |
We construct Feigenbaum quadratic polynomials whose Julia sets have positive Lebesgue measure. They provide first examples of rational maps for which the hyperbolic dimension is different from the Hausdorff dimension of the Julia set. The corresponding set of parameters has positive Hausdorff dimension.
We prove that if $\xi$ is a quasisymmetric homeomorphism between Sierpinski carpets that are the Julia sets of postcritically-finite rational maps, then $\xi$ is the restriction of a Mobius transformation to the Julia set. This implies that the group of quasisymmetric homeomorphisms of a Sierpinski carpet Julia set of a postcritically-finite rational map is finite.
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.
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.
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.
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).
In this paper we consider parabolic bifurcations of families of diffeomorphisms in two complex dimensions.
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.