Related papers: Discontinuity of straightening in anti-holomorphic…
It is well known that baby Mandelbrot sets are homeomorphic to the original one. We study baby Tricorns appearing in the Tricorn, which is the connectedness locus of quadratic anti-holomorphic polynomials, and show that the dynamically…
We continue the study of straightening maps for the family of polynomials of degree $d \ge 3$. The notion of straightening map is originally introduced by Douady and Hubbard to study relationship between polynomial-like renormalizations and…
We discuss self-similar property of the tricorn, the connectedness locus of the anti-holomorphic quadratic family. As a direct consequence of the study on straightening maps by Kiwi and the author, we show that there are many homeomorphic…
We describe a new algebro-geometric perspective on the study of the Milnor fibration and, as a first step toward putting it into practice, prove powerful criteria for a deformation of a holomorphic function germ to admit a stratification on…
The multicorns are the connectedness loci of unicritical antiholomorphic polynomials $\bar{z}^d + c$. We investigate the structure of boundaries of hyperbolic components: we prove that the structure of bifurcations from hyperbolic…
Renormalizations can be considered as building blocks of complex dynamical systems. This phenomenon has been widely studied for iterations of polynomials of one complex variable. Concerning non-polynomial hyperbolic rational maps, a recent…
The purpose of this paper is to initiate a theory concerning the dynamics of asymptotically holomorphic polynomial-like maps. Our maps arise naturally as deep renormalizations of asymptotically holomorphic extensions of $C^r$ ($r>3$)…
One of the conspicuous features of real slices of bicritical rational maps is the existence of Tricorn-type hyperbolic components. Such a hyperbolic component is called invisible if the non-bifurcating sub-arcs on its boundary do not…
We investigate the local dynamics of antiholomorphic diffeomorphisms around a parabolic fixed point. We first give a normal form. Then we give a complete classification including a modulus space for antiholomorphic germs with a parabolic…
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…
For Cartan geometries admitting automorphisms with isotropies satisfying a particular, loosely dynamical property on their model geometries, we demonstrate the existence of an open subset of the geometry with trivial holonomy. This…
Inou and Shishikura provided a class of maps that is invariant by near-parabolic renormalization, and that has proved extremely useful in the study of the dynamics of quadratic polynomials. We provide here another construction, using more…
The paper deals with dynamics of expanding Lorenz maps, which appear in a natural way as Poincar\`e maps in geometric models of well-known Lorenz attractor. Using both analytical and symbolic approaches, we study connections between…
In this revised version, applying a general renormalization procedure for formal self-maps, producing a formal normal form simpler than the classical Poincar\'e-Dulac normal form, we shall give a complete list of normal forms for…
We classify generic unfoldings of germs of antiholomorphic diffeomorphisms with a parabolic point of codimension~$k$ (i.e.~a fixed point of multiplicity $k+1$) under conjugacy. Such generic unfoldings depend real analytically on $k$ real…
The classical straightening theorem as proved by Douady and Hubbard shows that a polynomial-like sequence is hybrid equivalent to a polynomial. We generalize this result to non-autonomous iteration where one considers composition sequences…
We study the parameter space structure of degree $d \ge 3$ one complex variable polynomials as dynamical systems acting on $\C$. We introduce and study {\it straightening maps}. These maps are a natural higher degree generalization of the…
We study groups of germs of complex diffeomorphisms having a property called irreducibility. The notion is motivated by a similar property of the fundamental group of the complement of an irreducible hypersurface in the complex projective…
We study stability and bifurcations in holomorphic families of polynomial automorphisms of C^2. We say that such a family is weakly stable over some parameter domain if periodic orbits do not bifurcate there. We first show that this defines…
We give a way to construct group of pseudo-automorphisms of rational varieties of any dimension that fix pointwise the image of a cubic hypersurface of $P^n. These group are free products of involutions, and most of their elements have…