Related papers: Combinatorial structure of the parameter plane of …
The \emph{Cantor locus} is the unique hyperbolic component, in the moduli space of quadratic rational maps ${\bf rat}_2$, consisting of maps with totally disconnected Julia sets. Whereas the geometry and dynamics of the Cantor locus is well…
We introduce a generalization of the McMullen family $f_{\lambda}(z)=z^n+\lambda/z^d$. In 1988, C. McMullen showed that the Julia set of $f_{\lambda}$ is a Cantor set of circles if and only if $1/n+1/d<1$ and the simple critical values of…
We prove some new continuity results for the Julia sets $J$ and $J^{+}$ of the complex H\'enon map $H_{c,a}(x,y)=(x^{2}+c+ay, ax)$, where $a$ and $c$ are complex parameters. We look at the parameter space of dissipative H\'enon maps which…
In this article, we give a combinatorial model in terms of symmetric cores of the indexing set of the irreducible components of $\mathcal{H}_n^{\Gamma}$ (the $\Gamma$-fixed points of the Hilbert scheme of $n$ points in $\mathbb{C}^2$)…
We consider the bifurcation structure of one-parameter families of unimodal maps whose differentiability is only $C^1$. The structure of its bifurcation diagram can be a very wild one in such case. However we prove that in a certain…
We studied the parameter plane of the cosine functions with a fixed critical point. The hyperbolic components can be classified into three types: A, C and D. All the hyperbolic components are bounded and simply connected, except for the…
The fundamental group of the complement of a hyperplane arrangement in a complex vector space is an important topological invariant. The third rank of successive quotients in the lower central series of the fundamental group was called Falk…
We show the existence of computable complex numbers $\lambda$ for which the bifurcation locus of the one parameter complex family $f_{b}(z) = \lambda z + b z^{2} + z^{3}$ is not Turing computable.
We study the dynamics of polynomial maps on the boundary of the central hyperbolic component $\mathcal H_d$. We prove the local connectivity of Julia sets and a rigidity theorem for maps on the regular part of $\partial\mathcal H_d$. Our…
The hyperbolic components in the moduli space ${M}_d$ of degree $d\geq2$ rational maps are mysterious and fundamental topological objects. For those in the connectedness locus, they are known to be the finite quotients of the Euclidean…
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…
For an affine Lie algebra $\hat{\mathfrak g}$ the coefficients of certain vertex operators which annihilate level $k$ standard $\hat{\mathfrak g}$-modules are the defining relations for level $k$ standard modules. In the paper \cite{PS3}…
We explicitly describe a structure of a regular cell complex $K(L)$ on the moduli space $M(L)$ of a planar polygonal linkage $L$. The combinatorics is very much related (but not equal) to the combinatorics of the permutahedron. In…
We investigate the discontinuity of codings for the Julia set of a quadratic map. To each parameter ray, we associate a natural coding for Julia sets on the ray. Given a hyperbolic component $H$ of the Mandelbrot set, we consider the…
A non-elementary M\"obius group generated by two-parabolics is determined up to conjugation by one complex parameter and the parameter space has been extensively studied. In this paper, we use the results of \cite{GW} to obtain an…
We discuss the space of complex exponential maps $\Ek\colon z\mapsto e^{z}+\kappa$. We prove that every hyperbolic component $W$ has connected boundary, and there is a conformal isomorphism $\Phi_W\colon W\to\half^-$ which extends to a…
We study the family of singular perturbations of Blaschke products $B_{a,\lambda}(z)=z^3\frac{z-a}{1-\overline{a}z}+\frac{\lambda}{z^2}$. We analyse how the connectivity of the Fatou components varies as we move continuously the parameter…
Parabolic implosion describes the enrichment of Julia sets when a parabolic fixed point is perturbed. It is also natural to study parabolic implosion in parameter spaces. In particular, when one perturbs properly the family of cubic…
We prove that the hyperbolic components of bicritical rational maps having two distinct attracting cycles each of period at least two are bounded in the moduli space of bicritical rational maps. Our arguments rely on arithmetic methods.
The fundamental group of the complement of a hyperplane arrangement in a complex vector space is an important topological invariant. The third rank of successive quotients in the lower central series of the fundamental group was called Falk…