arXiv subjects
Publications and source records attributed to Javier Ribón.
We show recurrent phenomena for orbits of groups of local complex analytic diffeomorphisms that have a certain subgroup or image by a morphism of groups that is non-virtually solvable. In particular we prove that a non-virtually solvable subgroup of local biholomorphisms has always recurrent orbits, i.e. there exists an orbit contained in its set of limit points.
We prove that a nilpotent subgroup of orientation preserving $C^{1}$ diffeomorphisms of ${\mathbb S}^{2}$ has a finite orbit of cardinality at most two. We also prove that a finitely generated nilpotent subgroup of orientation preserving $C^{1}$ diffeomorphisms of ${\mathbb R}^{2}$ preserving a compact set has a global fixed point. These results generalize theorems of Franks, Handel and Parwani for the abelian case. We show that a nilpotent subgroup of orientation preserving $C^{1}$ diffeomorphisms of ${\mathbb S}^{2}$ that has a finite orbit of odd cardinality also has a global fixed point. Moreover we study the properties of the two-points orbits of nilpotent fixed-point-free subgroups of orientation preserving $C^{1}$ diffeomorphisms of ${\mathbb S}^{2}$.
We prove that a topological homeomorphism conjugating two generic 1-parameter unfoldings of 1-variable complex analytic resonant diffeomorphisms is holomorphic or anti-holomorphic by restriction to the unperturbed parameter. We provide examples that show that the genericity hypothesis is necessary. Moreover we characterize the possible behavior of conjugacies for the unperturbed parameter in the general case. In particular they are always real analytic outside of the origin. We describe the structure of the limits of orbits when we approach the unperturbed parameter. The proof of the rigidity results is based on the study of the action of a topological conjugation on the limits of orbits.
Let $G$ be a solvable subgroup of the group $\diff{}{n}$ of local complex analytic diffeomorphisms. Analogously as for groups of matrices we bound the solvable length of $G$ by a function of $n$. Moreover we provide the best possible bounds for connected, unipotent and nilpotent groups.
In [Xiang Zhang, The embedding flows of $C^{\infty}$ hyperbolic diffeomorphisms, J. Differential Equations 250 (2011), no. 5, 2283-2298] Zhang proved that any local smooth hyperbolic diffeomorphism whose eigenvalues are weakly nonresonant is embedded in the flow of a smooth vector field. We present a new, simpler and more conceptual proof of such result using the Jordan-Chevalley decomposition in algebraic groups and the properties of the exponential operator. We characterize the hyperbolic smooth (resp. formal) diffeomorphisms that are embedded in a smooth (resp. formal) flow. We introduce a criterium showing that the presence of weak resonances for a diffeomorphism plus two natural conditions imply that it is not embeddable. This solves a conjecture of Zhang. The criterium is optimal, we provide a method to construct embeddable diffeomorphisms with weak resonances if we remove any of the conditions.
We study the formal conjugacy properties of germs of complex analytic diffeomorphisms defined in the neighborhood of the origin of ${\mathbb C}^{n}$. More precisely, we are interested on the nature of formal conjugations along the fixed points set. We prove that there are formally conjugated local diffeomorphisms $ϕ, η$ such that every formal conjugation $\hatσ$ (i.e. $η\circ \hatσ = \hatσ \circ ϕ$) does not extend to the fixed points set $Fix (ϕ)$ of $ϕ$, meaning that it is not transversally formal (or semi-convergent) along $Fix (ϕ)$. We focus on unfoldings of 1-dimensional tangent to the identity diffeomorphisms. We identify the geometrical configurations preventing formal conjugations to extend to the fixed points set: roughly speaking, either the unperturbed fiber is singular or generic fibers contain multiple fixed points.
We prove the multisummability of the infinitesimal generator of unfoldings of finite codimension tangent to the identity 1-dimensional local complex analytic diffeomorphisms. We also prove the multisummability of Fatou coordinates and extensions of the Ecalle-Voronin invariants associated to these unfoldings. The quasi-analytic nature is related to the parameter variable. As an application we prove an isolated zeros theorem for the analytic conjugacy problem. The proof is based on good asymptotics of Fatou coordinates and the introduction of a new auxiliary tool, the so called multi-transversal flows. They provide the estimates and the combinatorics of sectors typically associated to summability. The methods are based on the study of the infinitesimal stability properties of the unfoldings.
The formal class of a germ of diffeomorphism $ϕ$ is embeddable in a flow if $ϕ$ is formally conjugated to the exponential of a germ of vector field. We prove that there are complex analytic unipotent germs of diffeomorphisms at $({\mathbb C}^{n},0)$ ($n>1$) 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 via potential theory.
We provide a complete system of invariants for the formal classification of complex analytic unipotent germs of diffeomorphism at $\cn{n}$ fixing the orbits of a regular vector field. We reduce the formal classification problem to solve a linear differential equation. Then we describe the formal invariants; their nature depends on the position of the fixed points set $Fix ϕ$ with respect to the regular vector field preserved by $ϕ$. We get invariants specifically attached to higher dimension ($n \geq 3$) although generically they are analogous to the one-dimensional ones.