On pinching deformations of rational maps
On rational maps with two critical points.
On the accessible points in the Julia sets of some entire functions
We prove that for some families of entire functions whose Julia set is the complement of the basin of attraction every branch of a tree of preimages starting from this basin is convergent.
On the Hausdorff dimension of a Julia set with a rationally indifferent periodic point
On the iteration of entire functions
On the spectrum of stochastic perturbations of the shift and Julia sets
We extend the Killeen-Taylor study [Nonlinearity 13 (2000)] by investigating in different Banach spaces (,c₀(ℕ),c(ℕ)) the point, continuous and residual spectra of stochastic perturbations of the shift operator associated to the stochastic adding machine in base 2 and in the Fibonacci base. For the base 2, the spectra are connected to the Julia set of a quadratic map. In the Fibonacci case, the spectrum is related to the Julia set of an endomorphism of ℂ².
One-sided Lebesgue Bernoulli maps of the sphere of degree and .
Polynomial diffeomorphisms of : VII. Hyperbolicity and external rays
Rational Misiurewicz maps for which the Julia set is not the whole sphere
We show that Misiurewicz maps for which the Julia set is not the whole sphere are Lebesgue density points of hyperbolic maps.
Real and imaginary parts of polynomial iterates.
Recurrence of entire transcendental functions with simple post-singular sets
We study how the orbits of the singularities of the inverse of a meromorphic function determine the dynamics on its Julia set, at least up to a set of (Lebesgue) measure zero. We concentrate on a family of entire transcendental functions with only finitely many singularities of the inverse, counting multiplicity, all of which either escape exponentially fast or are pre-periodic. For these functions we are able to decide whether the function is recurrent or not. In the case that the Julia set is...
Remarque sur les fonctions ayant le même ensemble de Julia
Scaling in a map of the two-torus.
Simultaneous reduction to normal forms of commuting singular vector fields with linear parts having Jordan blocks
We study the simultaneous linearizability of –actions (and the corresponding -dimensional Lie algebras) defined by commuting singular vector fields in fixing the origin with nontrivial Jordan blocks in the linear parts. We prove the analytic convergence of the formal linearizing transformations under a certain invariant geometric condition for the spectrum of vector fields generating a Lie algebra. If the condition fails and if we consider the situation where small denominators occur, then...
Smooth Gevrey normal forms of vector fields near a fixed point
We study germs of smooth vector fields in a neighborhood of a fixed point having an hyperbolic linear part at this point. It is well known that the “small divisors” are invisible either for the smooth linearization or normal form problem. We prove that this is completely different in the smooth Gevrey category. We prove that a germ of smooth -Gevrey vector field with an hyperbolic linear part admits a smooth -Gevrey transformation to a smooth -Gevrey normal form. The Gevrey order depends on...
Solving the sextic by iteration: a study in complex geometry and dynamics.
Some counterexamples in dynamics of rational semigroups.
Some open problems in higher dimensional complex analysis and complex dynamics.
We present a collection of problems in complex analysis and complex dynamics in several variables.
Strong almost reducibility for analytic and Gevrey quasi-periodic cocycles
This article is about almost reducibility of quasi-periodic cocycles with a diophantine frequency which are sufficiently close to a constant. Generalizing previous works by L.H. Eliasson, we show a strong version of almost reducibility for analytic and Gevrey cocycles, that is to say, almost reducibility where the change of variables is in an analytic or Gevrey class which is independent of how close to a constant the initial cocycle is conjugated. This implies a result of density, or quasi-density,...
Structure of the McMullen domain in the parameter planes for rational maps
We show that, for the family of functions where n ≥ 3 and λ ∈ ℂ, there is a unique McMullen domain in parameter space. A McMullen domain is a region where the Julia set of is homeomorphic to a Cantor set of circles. We also prove that this McMullen domain is a simply connected region in the plane that is bounded by a simple closed curve.