Fernando Sanz

Champs de vecteurs analytiques et champs de gradients

A theorem of Łojasiewicz asserts that any relatively compact solution of a real analytic gradient vector field has finite length. We show here a generalization of this result for relatively compact solutions of an analytic vector field X with a smooth invariant hypersurface, transversally hyperbolic for X, where the restriction of the field is a gradient. This solves some instances of R. Thoms Gradient Conjecture. Furthermore, if the dimension of the ambient space is three, these solutions do not oscillate (in the sense that they cut an analytic set only finitely many times); this can also be applied to some gradient vector fields.

ON RESTRICTED ANALYTIC GRADIENTS ON ANALYTIC ISOLATED SURFACE SINGULARITIES

Let (X,0) be a real analytic isolated surface singularity at the origin 0 of a real analytic manifold (R n ,0) equipped with a real analytic metric g. Given a real analytic function f0 : (R n ,0) ! (R,0) singular at 0, we prove that the gradient trajectories for the metric g|X\0 of the restriction (f0|X) escaping from or ending up at the origin 0 do not oscillate. Such a trajectory is thus a sub-pfaffian set. Moreover, in each connected component of X \ 0 where the restricted gradient does not vanish, there is always a trajectory accumulating at 0 and admitting a formal asymptotic expansion at 0.

Interlacing and Separation of Solutions of Linear Meromorphic ODEs

Solutions of two-dimensional linear systems of ODEs with real meromorphic coefficients may have two very distinct kinds of relative behaviour when they approach to a singular point: either any two of them are linked or either any two of them can be separated by a linear projection. In this paper, we are interesting in the question of the decidability of the dichotomy linked/separated for the whole family of systems. First, we rewrite the known result which asserts that the dichotomy is determined in terms of a semialgebraic set (is decidable) on a truncation of the Taylor expansion of the coefficients of the system. After that, we study the question of the decidability of that dichotomy in terms of the coefficients of the system themselves as elements of the ordered Hardy field of real meromorphic functions.

Balanced Coordinates for Spiraling Dynamics

We study some aspects of the dynamics of an analytic vector fieldX in a neighbourhood of an invariant non-singular curve τ in ℝ3. Namely, the spiraling behaviour: any trajectory ofX spirals asymptotically around τ. This is measured by means of the angle, θ, and the distance,r, functions of the trajectories with respect to cylindrical coordinates around τ. Coordinates are called balanced if these functions are monotone, which is not an intrinsic property. Balanced coordinates always exist in the case ofelementary singularity (non-nilpotent linear part) and we show, in the general case when τ is not contained in the singular locus ofX, the existence of coordinates for which the angle is monotone. These are obtained asmaximal contact coordinates for the reduction of the singularity. The results can be viewed as generalizations of the corresponding results in dimension two which we study first as a motivation.