Generalized Flow-Box property for singular foliations

We introduce a notion of generalized Flow-Box property valid for general singular distributions and sub-varieties (based on a dynamical interpretation). Just as in the usual Flow-Box Theorem, we characterize geometrical and algebraic conditions of (quasi) transversality in order for an analytic sub-variety $X$ (not necessarily regular) to be a section of a line foliation. We also discuss the case of more general foliations. This study is originally motivated by a question of Jean-Francois Mattei (concerning the strengthening of a Theorem of Mattei) about the existence of local slices for a (non-compact) Lie group action.Comment: Changes in Section

On the Nash points of subanalytic sets

Based on a recently developed rank Theorem for Eisenstein power series, we provide new proofs of the following two results of W. Pawlucki: I) The non regular locus of a complex or real analytic map is an analytic set. II) The set of semianalytic or Nash points of a subanalytic set X is a subanalytic set, whose complement has codimension two in X.Comment: Important: Our original pre-print arXiv:2205.03079 had two set of distinct results. We have divided that pre-print in two. This paper contains the second set of results ; v2 of the original submission contains the first set of results. We have divided our pre-prin

Topological classification of limit periodic sets of polynomial planar vector fields

We characterize the limit periodic sets of families of algebraic planar vector fields up to homeomorphisms. We show that any limit periodic set is topologically equivalent to a compact and connected semialgebraic set of the sphere of dimension 0 or 1. Conversely, we show that any compact and connected semialgebraic set of the sphere of dimension 0 or 1 can be realized as a limit periodic set

Bifurcation analysis of a system for population dynamics

Nesta dissertação, tratamos do estudo das bifurcações de um modelo bi-dimensional de presa-predador, que estende e aperfeiçoa o sistema de Lotka-Volterra. Tal modelo apresenta cinco parâmetros e uma função resposta não monotônica do tipo Holling IV: \\left\\{\\begin \\dot=x(1-\\lambda x-\\frac{\\alpha x^2+\\beta x +1})\\\\ \\dot=y(-\\delta-\\mu y+\\frac{\\alpha x^2+\\beta x +1}) \\end ight. Estudamos as bifurcações do tipo sela-nó, Hopf, transcrítica, Bogdanov-Takens e Bogdanov-Takens degenerada. O método dos centros organizadores é usado para estudar o comportamento qualitativo do diagrama de bifurcação.In this work are studied the bifurcations of a bi-dimensional predator-prey model, which extends and improves the Volterra-Lotka system. This model has five parameters and a non-monotonic response function of Holling IV type: \\left\\{\\begin \\dot=x(1-\\lambda x-\\frac{\\alpha x^2+\\beta x +1})\\\\ \\dot=y(-\\delta-\\mu y+\\frac{\\alpha x^2+\\beta x +1}) \\end ight. They studied the sadle-node, Hopf, transcritic, Bogdanov-Takens and degenerate Bogdanov-Takens bifurcations. The method of organising centers is used to study the qualitative behavior of the bifurcation diagram

: École d’Été 2019 - Feuilletages et géométrie algébrique

Given a totally nonholonomic distribution of rank two $\Delta$ on a three-dimensional manifold $M$, it is natural to investigate the size of the set of points $\mathcal{X}^x$ that can be reached by singular horizontal paths starting from a same point $x \in M$. In this setting, the Sard conjecture states that $\mathcal{X}^x$ should be a subset of the so-called Martinet surface of 2-dimensional Hausdorff measure zero.I will present a reformulation of the conjecture in terms of the behavior of a (real) singular foliation. Next, I will present a recent work in collaboration with A. Figalli, L. Rifford and A. Parusinski, where we show that the (strong version of the) conjecture holds in the analytic category and in dimension 3. Our methods rely on resolution of singularities of surfaces, foliations and metrics; regularity analysis of Poincaré transition maps; and on a simpletic argument, concerning a transversal metric of an isotropic singular foliation

INNER GEOMETRY OF COMPLEX SURFACES: A VALUATIVE APPROACH

Given a complex analytic germ (X, 0) in (C n , 0), the standard Hermitian metric of C n induces a natural arc-length metric on (X, 0), called the inner metric. We study the inner metric structure of the germ of an isolated complex surface singularity (X, 0) by means of an infinite family of numerical analytic invariants, called inner rates. Our main result is a formula for the Laplacian of the inner rate function on a space of valuations, the non-archimedean link of (X, 0). We deduce in particular that the global data consisting of the topology of (X, 0), together with the configuration of a generic hyperplane section and of the polar curve of a generic plane projection of (X, 0), completely determine all the inner rates on (X, 0), and hence the local metric structure of the germ. Several other applications of our formula are discussed in the paper

: École d’Été 2019 - Feuilletages et géométrie algébrique

Given a totally nonholonomic distribution of rank two $\Delta$ on a three-dimensional manifold $M$, it is natural to investigate the size of the set of points $\mathcal{X}^x$ that can be reached by singular horizontal paths starting from a same point $x \in M$. In this setting, the Sard conjecture states that $\mathcal{X}^x$ should be a subset of the so-called Martinet surface of 2-dimensional Hausdorff measure zero.I will present a reformulation of the conjecture in terms of the behavior of a (real) singular foliation. Next, I will present a recent work in collaboration with A. Figalli, L. Rifford and A. Parusinski, where we show that the (strong version of the) conjecture holds in the analytic category and in dimension 3. Our methods rely on resolution of singularities of surfaces, foliations and metrics; regularity analysis of Poincaré transition maps; and on a simpletic argument, concerning a transversal metric of an isotropic singular foliation

INNER GEOMETRY OF COMPLEX SURFACES: A VALUATIVE APPROACH

International audienceGiven a complex analytic germ (X, 0) in (C n , 0), the standard Hermitian metric of C n induces a natural arc-length metric on (X, 0), called the inner metric. We study the inner metric structure of the germ of an isolated complex surface singularity (X, 0) by means of an infinite family of numerical analytic invariants, called inner rates. Our main result is a formula for the Laplacian of the inner rate function on a space of valuations, the non-archimedean link of (X, 0). We deduce in particular that the global data consisting of the topology of (X, 0), together with the configuration of a generic hyperplane section and of the polar curve of a generic plane projection of (X, 0), completely determine all the inner rates on (X, 0), and hence the local metric structure of the germ. Several other applications of our formula are discussed in the paper