• Corpus ID: 119662986

# Julia sets appear quasiconformally in the Mandelbrot set

@article{Kawahira2018JuliaSA,
title={Julia sets appear quasiconformally in the Mandelbrot set},
author={Tomoki Kawahira and Masashi Kisaka},
journal={arXiv: Dynamical Systems},
year={2018}
}
• Published 31 March 2018
• Mathematics
• arXiv: Dynamical Systems
In this paper we prove the following: Take any "small Mandelbrot set" and zoom in a neighborhood of a parabolic or Misiurewicz parameter in it, then we can see a quasiconformal image of a Cantor Julia set which is a perturbation of a parabolic or Misiurewicz Julia set. Furthermore, zoom in its middle part, then we can see a certain nested structure ("decoration") and finally another "smaller Mandelbrot set" appears. A similar nested structure exists in the Julia set for any parameter in the…

## Figures from this paper

Renormalization and embedded Julia sets in the Mandelbrot set
The decorations of a small Mandelbrot set within the Mandelbrot set M contain embedded Julia sets; these are Cantor sets quasiconformally homeomorphic to a quadratic Julia set. So the local geometry

## References

SHOWING 1-10 OF 53 REFERENCES
Similarity Between the Mandelbrot Set and Julia Sets
The Mandelbrot set M is "self-similar" about any Misiurewicz point c in the sense that if we examine a neighborhood of c in M with a very powerful microscope, and then increase the magnification by a
On the continuity of Hausdorff dimension of Julia sets and similarity between the Mandelbrot set and Julia sets
Given d ≥ 2 consider the family of polynomials Pc(z) = z + c for c ∈ C. Denote by Jc the Julia set of Pc and letMd = {c | Jc is connected} be the connectedness locus; for d = 2 it is called the
On the Lebesgue measure of the Julia set of a quadratic polynomial
The goal of this note is to prove the following theorem: Let \$p_a(z) = z^2+a\$ be a quadratic polynomial which has no irrational indifferent periodic points, and is not infinitely renormalizable. Then
Periodic Orbits, Externals Rays and the Mandelbrot Set: An Expository Account
A key point in Douady and Hubbard's study of the Mandelbrot set \$M\$ is the theorem that every parabolic point \$c\ne 1/4\$ in \$M\$ is the landing point for exactly two external rays with angle which are
Similarity between the Mandelbrot set and Julia sets
The Mandelbrot setM is “self-similar” about any Misiurewicz pointc in the sense that if we examine a neighborhood ofc inM with a very powerful microscope, and then increase the magnification by a
Julia Sets in Parameter Spaces
• Mathematics
• 2001
Abstract: Given a complex number λ of modulus 1, we show that the bifurcation locus of the one parameter family {fb(z)=λz+bz2+z3}b∈ℂ contains quasi-conformal copies of the quadratic Julia set
Laminations in holomorphic dynamics
• Mathematics
• 1994
We suggest a way to associate to a rational map of the Riemann sphere a three dimensional object called a hyperbolic orbifold 3-lamination. The relation of this object to the map is analogous to the
On a theorem of Fatou
We prove a result on the backward dynamics of a rational function nearby a point not contained in the ω-limit set of a recurrent critical point. As a corollary we show that a compact invariant subset
The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets
It is shown that the boundary of the Mandelbrot set M has Hausdorff dimension two and that for a generic c E AM, the Julia set of z I > Z2 + C also has Hausdorff dimension two. The proof is based on
Tessellation and Lyubich–Minsky laminations associated with quadratic maps, I: pinching semiconjugacies
Abstract We construct tessellations of the filled Julia sets of hyperbolic and parabolic quadratic maps. The dynamics inside the Julia sets are then organized by tiles which play the role of the