Antiholomorphic Polynomials Research Articles

Discontinuity of Straightening in Anti-holomorphic Dynamics: I

It is well known that baby Mandelbrot sets are homeomorphic to the original one. We study baby Tricorns appearing in the Tricorn, which is the connectedness locus of quadratic anti-holomorphic polynomials, and show that the dynamically natural straightening map from a baby Tricorn to the original Tricorn is discontinuous at infinitely many explicit parameters. This is the first known example of discontinuity of straightening maps on a real two-dimensional slice of an analytic family of holomorphic polynomials. The proof of discontinuity is carried out by showing that all non-real umbilical cords of the Tricorn wiggle, which settles a conjecture made by various people including Hubbard, Milnor, and Schleicher.

Univalent polynomials and Hubbard trees

We study rational functions $f$ of degree $d+1$ such that $f$ is univalent in the exterior unit disc, and the image of the unit circle under $f$ has the maximal number of cusps ($d+1$) and double points $(d-2)$. We introduce a bi-angled tree associated to any such $f$. It is proven that any bi-angled tree is realizable by such an $f$, and moreover, $f$ is essentially uniquely determined by its associated bi-angled tree. This combinatorial classification is used to show that such $f$ are in natural 1:1 correspondence with anti-holomorphic polynomials of degree $d$ with $d-1$ distinct, fixed critical points (classified by their Hubbard trees).

Discontinuity of Straightening in Anti-Holomorphic Dynamics: II

Abstract In [21], Milnor found Tricorn-like sets in the parameter space of real cubic polynomials. We give a rigorous definition of these Tricorn-like sets as suitable renormalization loci and show that the dynamically natural straightening map from such a Tricorn-like set to the original Tricorn is discontinuous. We also prove some rigidity theorems for polynomial parabolic germs, which state that one can recover unicritical holomorphic and anti-holomorphic polynomials from their parabolic germs.

CR Iteration in Generation of Antifractals With s-Convexity

Complex graphics of nonlinear dynamical systems perform vital role in many fields, e.g., image compression or encryption, art, science, and so on. Antifractals are generated by applying a map recursively to an initial point in complex plane that have become a significant area of research these days. The purpose of this paper is to investigate the dynamics of antifractals like anti-Julia sets, tricornsand multicorns of antiholomorphic polynomials via CR iteration scheme with s-convexity. Many beautiful aesthetic patterns are visualized for antipolynomial z → z m + c of complex polynomial z m + c, for m ≥ 2 to explore the geometry of antifractals.

On multicorns and unicorns II: bifurcations in spaces of antiholomorphic polynomials

Themulticornsare the connectedness loci of unicritical antiholomorphic polynomials$\bar{z}^{d}+c$. 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.

Non-landing parameter rays of the multicorns

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 $\mathcal{M}_d^*$, 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 $\mathcal{M}_d^*$, and the Misiurewicz parameters are not dense on the boundary of $\mathcal{M}_d^*$.

Orbit portraits of unicritical antiholomorphic polynomials

Orbit portraits were introduced by Goldberg and Milnor as a combinatorial tool to describe the patterns of all periodic dynamical rays landing on a periodic cycle of a quadratic polynomial. This encodes information about the dynamics and the parameter spaces of these maps. We carry out a similar analysis for unicritical antiholomorphic polynomials, and give an explicit description of the orbit portraits that can occur for such maps in terms of their characteristic angles, which turns out to be rather restricted when compared with the holomorphic case. Finally, we prove a realization theorem for these combinatorial objects. The results obtained in this paper serve as a combinatorial foundation for a detailed understanding of the combinatorics and topology of the parameter spaces of unicritical antiholomorphic polynomials and their connectedness loci, known as the multicorns.

Coherent state wave functions on a torus with a constant magnetic field

We study two alternative definitions of localized states in the lowest Landau level (LLL) on a torus. One definition is to construct localized states, as the projection of the coordinate delta function onto the LLL. Another definition, proposed by Haldane, is to consider the set of functions which have all their zeros at a single point. Since a LLL wave function on a torus, supporting Nϕ magnetic flux quanta, is uniquely defined by the position of its Nϕ zeros, this defines a set of functions that are expected to be localized around the point maximally far away from the zeros. These two families of localized states have many properties in common with the coherent states on the plane and on the sphere, viz a resolution of unity and a self-reproducing kernel. However, we show that only the projected delta function is maximally localized. Additionally, we show how to project onto the LLL, functions that contain holomorphic derivatives and/or anti-holomorphic polynomials, and apply our methods in the description of hierarchical quantum Hall liquids.

ON MULTICORNS AND UNICORNS I: ANTIHOLOMORPHIC DYNAMICS, HYPERBOLIC COMPONENTS AND REAL CUBIC POLYNOMIALS

We investigate the dynamics and the bifurcation diagrams of iterated antiholomorphic polynomials: These are complex conjugates of ordinary polynomials. Their second iterates are holomorphic polynomials, but dependence on parameters is only real-analytic. The structure of hyperbolic components of the family of unicritical antiholomorphic polynomials is revealed. In case of degree two, they arise naturally in the parameter space of real cubic (holomorphic) polynomials, which we investigate as well.

Homogenization, Reflection, and the X-variety

The author uses a homogenization technique to analyze a reflection principle in complex analysis. Given a holomorphic polynomial on complex Euclidean space, the author introduces its reflection, which is a matrix of anti-holomorphic polynomials, and establishes basic properties. The introduction to the paper discusses the classical one variable case, where the reflection is well known. The main result shows that the X-variety, defined by Forstneric in his study of proper mappings between balls, is an affine space determined by the null space of the reflection matrix. An integral formula for this matrix is also given. The author provides examples both when the X-variety is a bundle and when it is not a bundle, and then proves several general results to this effect. The final result gives a new family of examples where the X-variety of a map equals the graph of the map.