Related papers: Shapes of Polynomial Julia Sets

A generalization of the filled-in Julia set is presented using the multicomplex numbers and an algorithm is presented to visualize these sets in the tridimensional space. There are many ways to visualize these higher dimensional fractals…

We investigate the approximate j-dimensionality of the singularity sets of minimal surfaces prescribed by Simon. This leads to the clasification of 8 variations of approximately j-dimensional surfacs in terms of dimension and locally finite…

Let $f(z) = z^2 + c$ be a quadratic polynomial, with c in the Mandelbrot set. Assume further that both fixed points of f are repelling, and that f is not renormalizable. Then we prove that the Julia set J of f is holomorphically removable…

The matching problem for a given Jordan curve in the complex plane asks to find two nonconstant functions, one analytic in the bounded complementary component of the curve and the other analytic in the unbounded complementary component of…

There have been several propositions for a geometric and essentially non-linear formulation of quantum mechanics. From a purely mathematical point of view, the point of view of Jordan algebra theory might give new strength to such…

We elucidate the geometry of matrix models based on simple formally real Jordan algebras. Such Jordan algebras give rise to a nonassociative geometry that is a generalization of Lorentzian geometry. We emphasize constructions for the…

It is shown that for quasiregular maps of positive lower order the Julia set coincides with the boundary of the fast escaping set.

We study the functional equation $A\circ X=X\circ B$, where $A,$ $B$, and $X$ are polynomials over $\mathbb C$. Using previous results of the author about polynomials sharing preimages of compact sets, we show that for given $B$ its…

We look for minimal conditions on a two-dimensional metric surface $X$ of locally finite Hausdorff $2$-measure under which $X$ admits an (almost) parametrization with good geometric and analytic properties. Only assuming that $X$ is locally…

We give three families of parabolic rational maps and show that every Cantor set of circles as the Julia set of a non-hyperbolic rational map must be quasisymmetrically equivalent to the Julia set of one map in these families for suitable…

We investigate the dynamics of semigroups generated by a family of polynomial maps on the Riemann sphere such that the postcritical set in the complex plane is bounded. The Julia set of such a semigroup may not be connected in general. We…

We give a classification of maximal elements of the set of finite groups that can be realized as the full automorphism groups of simple polarized abelian fourfolds over finite fields. As an application, we compute the Jordan constants of…

Jordan algebras arise naturally in (quantum) information geometry, and we want to understand their role and their structure within that framework. Inspired by Kirillov's discussion of the symplectic structure on coadjoint orbits, we provide…

We survey the definition of the radial Julia set of a meromorphic function (in fact, more generally, any "Ahlfors islands map"), and give a simple proof that the Hausdorff dimension of the reduced Julia set always coincides with the…

This paper presents some results on simple exceptional Jordan algebra over algebraically closed field $\Phi$ with characteristic not 2. Namely an example of simple decomposition of $H(O_3)$ into the sum of two subalgebras of the type…

We complete the enumeration of the possible roots of the Alexander polynomial (both conventional and over finite fields) of a trigonal curve. The curves are not assumed proper or irreducible.

We discuss computability and computational complexity of conformal mappings and their boundary extensions. As applications, we review the state of the art regarding computability and complexity of Julia sets, their invariant measures and…

If the preimage of a four-point set under a meromorphic function belongs to the real line, then the image of the real line is contained in a circle in the Riemann sphere. We include an application of this result to holomorphic dynamics: if…

We show that, when a non-integrable rational map changes to an integrable one continuously, a large part of the Julia set of the map approach indeterminate points (IDP) of the map along algebraic curves. We will see that the IDPs are…

We extend results about the dimension of the radial Julia set of certain exponential functions to quasiregular Zorich maps in higher dimensions. Our results improve on previous estimates of the dimension also in the special case of…