相关论文: A new geometric characterization of the Julia set

It is shown that the boundary of the Mandelbrot set $M$ has Hausdorff dimension two and that for a generic $c \in \bM$, the Julia set of $z \mapsto z^2+c$ also has Hausdorff dimension two. The proof is based on the study of the bifurcation…

We construct a geometrically self-similar Cantor set $X$ of genus $2$ in $\mathbb{R}^3$. This construction is the first for which the local genus is shown to be $2$ at every point of $X$. As an application, we construct, also for the first…

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…

A. Sannami constructed an example of the differentiable Cantor set embedded in the real line whose difference set has a positive measure. In this paper, we generalize the definition of the difference sets for sets of the two dimensional…

Fractals offer the ability to generate fascinating geometric shapes with all sorts of unique characteristics (for instance, fractal geometry provides a basis for modelling infinite detail found in nature). While fractals are non-euclidean…

We show that the Hausdorff dimension of the set of points of bounded orbit in the Julia set of a meromorphic map with a simply connected direct tract and a certain restriction on the singular values is strictly greater than one. This result…

This paper describes new results on the growth and zeros of the Ruelle zeta function for the Julia set of a hyperbolic rational map. It is shown that the zeta function is bounded by $\exp(C_K |s|^{\delta})$ in strips $|\Re s| \leq K$, where…

Julia and Mandelbrot sets, which characterize bounded orbits in dynamical systems over the complex numbers, are classic examples of fractal sets. We investigate the analogs of these sets for dynamical systems over the hyperbolic numbers.…

We study rigidity of rational maps that come from Newton's root finding method for polynomials of arbitrary degrees. We establish dynamical rigidity of these maps: each point in the Julia set of a Newton map is either rigid (i.e. its orbit…

This paper deals with a new notion called fuzzy $\alpha$-cut and its properties. A notion called localic frame is also introduced. Algebraic structures arising out of the family of fuzzy $\alpha$-cuts have been investigated. It will be seen…

We study topological properties of the escaping endpoints and fast escaping endpoints of the Julia set of complex exponential $\exp(z)+a$ when $a\in (-\infty,-1)$. We show neither space is homeomorphic to the whole set of endpoints. This…

In the famous work by Buff and Ch\'eritat constructing quadratic Julia sets with positive area, the control of the shape of perturbed Siegel disks is a key technique. To do it, Buff and Ch\'eritat added a restrictive condition on rotation…

This article deals with the question of local connectivity of the Julia set of polynomials and rational maps. It essentially presents conjectures and questions.

The Fatou-Julia iteration theory of rational functions has been extended to quasiregular mappings in higher dimension by various authors. The purpose of this paper is an analogous extension of the iteration theory of transcendental entire…

We explore the connected/disconnected dichotomy for the Julia set of polynomial automorphisms of C^2. We develop several aspects of the question, which was first studied by Bedford-Smillie. We introduce a new sufficient condition for the…

We prove that a polynomial Julia set which is a finitely irreducible continuum is either an arc or an indecomposable continuum. For the more general case of rational functions, we give a topological model for the dynamics when the Julia set…

Sun Daochun and Yang Lo have shown that many results of the Fatou-Julia iteration theory of rational functions extend to quasiregular self-maps of the Riemann sphere for which the degree exceeds the dilatation. We show that in this context,…

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…

According to the Thurston No Wandering Triangle Theorem, a branching point in a locally connected quadratic Julia set is either preperiodic or precritical. Blokh and Oversteegen proved that this theorem does not hold for higher degree Julia…

We prove that if $\xi$ is a quasisymmetric homeomorphism between Sierpi\'nski carpets that are the Julia sets of postcritically-finite rational maps, then $\xi$ is the restriction of a M\"obius transformation to the Julia set. This implies…