Related papers: Combinatorial models for spaces of cubic polynomia…
For the study of the 2-dimensional space of cubic polynomials, J. Milnor considers the complex 1-dimensional slice S_n of the cubic polynomials which have a super-attracting orbit of period n. He gives in [M4] a detailed conjectural picture…
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.…
Thurston introduced $\si_d$-invariant laminations (where $\si_d(z)$ coincides with $z^d:\ucirc\to \ucirc$, $d\ge 2$) and defined \emph{wandering $k$-gons} as sets $\T\subset \ucirc$ such that $\si_d^n(\T)$ consists of $k\ge 3$ distinct…
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 study asymptotic dynamics in networks of coupled quadratic nodes. While single map complex quadratic iterations have been studied over the past century, considering ensembles of such functions, organized as coupled nodes in a network,…
We prove an extension results for the multiplier of an attracting periodic orbit of a quadratic map as a function of the parameter. This has applications to the problem of geometry of the Mandelbrot and Julia sets. In particular, we prove…
Hyperbolic numbers are a variation of complex numbers, but their dynamics is quite different. The hyperbolic Mandelbrot set for quadratic functions over hyperbolic numbers is simply a filled square, and the filled Julia set for hyperbolic…
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…
We continue the study of constructing invariant Laplacians on Julia sets, and studying properties of their spectra. In this paper we focus on two types of examples: 1) Julia sets of cubic polynomials $z^3 + c$ with a single critical point;…
We describe a locally connected model of the boundary of the cubic connectedness locus. The model is obtained by constructing a decomposition of the space of critical portraits and collapsing elements of the decomposition into points. This…
We describe a primary limb structure in the connectedness locus of complex cubic polynomials, where the limbs are indexed by the periodic points of the doubling map $t \mapsto 2t \ (\operatorname{mod} {\mathbb Z})$. The main renormalization…
We give new proofs that the Mandelbrot set is locally connected at every Misiurewicz point and at every point on the boundary of a hyperbolic component. The idea is to show ``shrinking of puzzle pieces'' without using specific puzzles.…
Given a critically periodic quadratic map with no secondary renormalizations, we introduce the notion of $Q$-recurrent quadratic polynomials. We show that the pieces of the principal nest of a $Q$-recurrent map $f_c$ converge in shape to…
One of the main questions in the field of complex dynamics is the question whether the Mandelbrot set is locally connected, and related to this, for which maps the Julia set is locally connected. In this paper we shall prove the following…
For an infinitely renormalizable quadratic map $f_c: z\mapsto z^2+c$ with the sequence of renormalization periods ${k_m}$ and rotation numbers ${t_m=p_m/q_m}, we prove that if $\limsup k_m^{-1}\log |p_m|>0$, then the Mandelbrot set is…
One of the fundamental properties of the Mandelbrot set is that the set of postcritically finite parameters is structured like a tree. We extend this result to the set of quadratic kneading sequences and show that this space contains no…
We prove that topologically conjugate non-renormalizable polynomials are quasi-conformally conjugate. From this we derive that each such polynomial can be approximated by a hyperbolic polynomial. As a by product we prove that the Julia set…
We prove combinatorial rigidity of infinitely renormalizable unicritical polynomials, P_c :z \mapsto z^d+c, with complex c, under the a priori bounds and a certain "combinatorial condition". This implies the local connectivity of the…
The Mandelbrot set is a fractal which classifies the behaviour of complex quadratic polynomials. Although its remarkably simple definition: $\mathcal{M}:=\{c \in \mathbb{C}\,|\,Q_c(0)^n \nrightarrow \infty \mbox{ as } n\rightarrow \infty,…
Given $p/q$ and $p'/q$ both irreducible, we construct homeomorphisms between the $p/q$ and the $p'/q$ limbs of the Mandelbrot set. This homeomorphisms are not compatible with the dynamics. Moreover, the filled Julia sets of corresponding…