The Mandelbrot set is universal

Extension of the Douady-Hubbard's Theorem on Connectedness of the Mandelbrot Set to Symmetric Polynimials

Higher bifurcation currents, neutral cycles and the Mandelbrot set

We prove that given any θ1, . . . , θ2d−2 ∈ R\Z, the support of the bifurcation measure of the moduli space of degree d rational maps coincides with the closure of classes of maps having 2d − 2 neutral cycles of respective multipliers e1 , . . . , e2d−2 . To this end, we generalize a famous result of McMullen, proving that homeomorphic copies of (∂M) are dense in the support of the k-bifurcatio...

Inaccessibility and undecidability in computation, geometry, and dynamical systems

Non-self-similar sets defined by a decision procedure are numerically investigated by introducing the notion of inaccessibility to (ideal) decision procedure, that is connected with undecidability. A halting set of a universal Turing machine (UTM), the Mandelbrot set and a riddled basin are mainly investigated as non-self-similar sets with a decision procedure. By encoding a symbol sequence to ...

Multifractal Spectrum and Thermodynamical Formalism of the Farey Tree

Let (Ω, μ) be a set of real numbers to which we associate a measure μ. Let α ≥ 0, let Ωα = {x ∈ Ω/α(x) = α}, where α is the concentration index defined by Halsey et al. [Halsey et al., 1986]. Let fH(α) be the Hausdorff dimension of Ωα. Let fL(α) be the Legendre spectrum of Ω, as defined in [Riedi and Mandelbrot, 1998]; and fC(α) the classical computational spectrum of Ω, defined in [Halsey et a...

A Generalized Mandelbrot Set Based On Distance Ratio

The iteration of complex function can generate beautiful fractal images. This paper presents a novel method based on the iteration of the distance ratio with two points, which generates a generalized Mandelbrot set according to distance ratio convergence times. This paper states the definition of distance ratio and its iteration. Then taking the complex function f(z)=z+c for example, it discuss...