Related papers: Parabolic implosion in dimension 2

Starting from a hyperbolic toral automorphism, we obtain, for a small volume preserving perturbation, an exact and rigorous second order perturbation expansion of the Lyapunov exponents.

We show that there exist polynomial endomorphisms of C^2, possessing a wandering Fatou component. These mappings are polynomial skew-products, and can be chosen to extend holomorphically of P^2(C). We also find real examples with wandering…

How topological defects affect the dynamics of particles hopping between lattice sites of a distorted, two-dimensional crystal is addressed. Perturbation theory and numerical simulations show that weak, short-ranged topological disorder…

A discontinuous change in the size of an attractor is the most easily observed type of global bifurcation. More generally, an explosion is a discontinuous change in the set of recurrent points. An explosion often results from heteroclinic…

In this paper we study the geometry of the attractors of holomorphic maps with an irrationally indifferent fixed point. We prove that for an open set of such holomorphic systems, the local attractor at the fixed point has Hausdorff…

Consider the parameter space $\mathcal{P}_{\lambda}\subset \mathbb{C}^{2}$ of complex H\'enon maps $$ H_{c,a}(x,y)=(x^{2}+c+ay,ax),\ \ a\neq 0 $$ which have a semi-parabolic fixed point with one eigenvalue $\lambda=e^{2\pi i p/q}$. We give…

A theorem of Picard's type is proved for entire holomorphic mappings into complex projective varieties. This theorem has local character in the sense that the existence of Julia directions can be proved under a natural additional…

Analysis of the periodic points of a conservative periodic dynamical system uncovers the basic kinematic structure of the transport dynamics, and identifies regions of local stability or chaos. While elliptic and hyperbolic points typically…

Many authors have studied the dynamics of hyperbolic transcendental entire functions; these are those for which the postsingular set is a compact subset of the Fatou set. Equivalenty, they are characterized as being expanding.…

We prove that for each characteristic direction $[v]$ of a tangent to the identity diffeomorphism of order $k+1$ in $\mathbb{C}^2$ there exist either an analytic curve of fixed points tangent to $[v]$ or $k$ parabolic manifolds where all…

It has been known for a long time that the equivariant 2+1 wave map into the 2-sphere blows up if the initial data are chosen appropriately. Here, we present numerical evidence for the stability of the blow-up phenomenon under explicit…

We study bifurcation for the constant scalar curvature equation along a one-parameter family of Riemannian metrics on the total space of a harmonic Riemannian submersion. We provide an existence theorem for bifurcation points and a…

We analyze the infinitesimal effect of holomorphic perturbations of the dynamics of a structurally stable rational map on a neighborhood of its Julia set. This implies some restrictions on the behavior of critical points.

In wall-bounded parallel flows, sustained turbulence can occur even while laminar flow is still stable. Channel flow is one of such flows and displays spatio-temporal fluctuating patterns of localized turbulence along its way from/to…

For the quadratic family $f_{c}(z) = z^2+c$ with $c$ in a hyperbolic component of the Mandelbrot set, it is known that every point in the Julia set moves holomorphically. In this paper we give a uniform derivative estimate of such a motion…

The Leau-Fatou flower theorem completely describes the dynamic behavior of $1-$dimensional maps tangent to the identity. In dimension two Hakim and Abate proved that if $f$ is a holomorphic map tangent to the identity in $\mathbb{C}^2$ and…

We prove that if $F$ is a tangent to the identity diffeomorphism at $0\in\mathbb{C}^2$ and $\Gamma$ is a formal invariant curve of $F$ then there exists a parabolic curve (attracting or repelling) of $F$ asymptotic to $\Gamma$. The result…

Let $g$ be a polynomial automorphism of $\C^2$. We study the Hausdorff dimension and topological dimension of the Julia set of $g$. We show that when $g$ is a hyperbolic mapping, then the Hausdorff dimension of the Julia set is strictly…

For a hyperbolic polynomial automorphism of C^2 with a disconnected Julia set, and under a mild dissipativity condition, we give a topological description of the components of the Julia set. Namely, there are finitely many "quasi-solenoids"…

In this paper we describe the bifurcation diagram of the$2$-parameter family of vector fields $\dot z = z(z^k+\epsilon_1z+\epsilon_0)$ over $\mathbb C\mathbb P^1$ for $(\epsilon_1,\epsilon_0)\in \mathbb C^2$. There are two kinds of…