The formal class of a germ of diffeomorphism $\varphi$ is embeddable in a flow if $\varphi$ is formally conjugated to the exponential of a germ of vector field. We prove that there are complex analytic unipotent germs of diffeomorphisms at ${ℂ}^n$ ($n\>1$) whose formal class is non-embeddable. The examples are inside a family in which the non-embeddability is of geometrical type. The proof relies on the properties of some linear functional operators that we obtain through the study of polynomial families of diffeomorphisms...

We show that for entire maps of the form z ↦ λexp(z) such that the orbit of zero is bounded and Lebesgue almost every point is transitive, no absolutely continuous invariant probability measure can exist. This answers a long-standing open problem.

We consider the family of transcendental entire maps given by $f_a\left(z\right)=a(z-(1-a))exp(z+a)$ where a is a complex parameter. Every map has a superattracting fixed point at z = -a and an asymptotic value at z = 0. For a > 1 the Julia set of $f_a$ is known to be homeomorphic to the Sierpiński universal curve, thus containing embedded copies of any one-dimensional plane continuum. In this paper we study subcontinua of the Julia set that can be defined in a combinatorial manner. In particular, we show the existence of non-landing...

We consider a transcendental meromorphic function f belonging to the class ℬ (with bounded set of singular values). We show that if the Julia set J(f) is the whole complex plane ℂ, and the closure of the postcritical set P(f) is contained in B(0,R) ∪ {∞} and is disjoint from the set Crit(f) of critical points, then every compact and forward invariant set is hyperbolic, provided that it is disjoint from Crit(f). It is further shown, under general additional hypotheses, that f admits no measurable...

In this article, we study germs of holomorphic vector fields which are “higher order” perturbations of a quasihomogeneous vector field in a neighborhood of the origin of ${ℂ}^n$, fixed point of the vector fields. We define a “Diophantine condition” on the quasihomogeneous initial part $S$ which ensures that if such a perturbation of $S$ is formally conjugate to $S$ then it is also holomorphically conjugate to it. We study the normal form problem relatively to $S$. We give a condition on $S$ that ensures that there...

Let Γ be a finite subgroup of GL(n, C). This subgroup acts on the space of germs of holomorphic vector fields vanishing at the origin in Cn and on the group of germs of holomorphic diffeomorphisms of (Cn, 0). We prove a theorem of invariant conjugacy to a normal form and linearization for the subspace of invariant germs of holomorphic vector fields and we give a description of this type of normal forms in dimension n = 2.

We establish a Poincaré-Dulac theorem for sequences ${\left(G_n\right)}_{n\in \mathbb{Z}}$ of holomorphic contractions whose differentials $d_0{G}_n$ split regularly. The resonant relations determining the normal forms hold on the moduli of the exponential rates of contraction. Our results are actually stated in the framework of bundle maps.Such sequences of holomorphic contractions appear naturally as iterated inverse branches of endomorphisms of $ℂ{\mathbb{P}}^k$. In this context, our normalization result allows to estimate precisely the distortions of ellipsoids...

We present a geometric proof of the Poincaré-Dulac Normalization Theorem for analytic vector fields with singularities of Poincaré type. Our approach allows us to relate the size of the convergence domain of the linearizing transformation to the geometry of the complex foliation associated to the vector field.