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2 Bulletin of the Polish Academy of Sciences. Mathematics

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The limit directions of a real analytic vector field

Elkhadiri A.

Bulletin of the Polish Academy of Sciences. Mathematics

2000

Vol. 48, no 2

171-185

To every real analytic manifold M there is associate the so-called projective analytic tangent bundle, q : P(TM) -> M, whose fibers [q^-1](x) are the real projective spaces, P(T[sub x]M), of the tangent vector space T[sub x]M of M at x. Let v be a real analytic vector field on M. Let v be the section of q on the open set U =[...], defined by v(x) = Rv(x), for all x [belongs to] U. The closure of Im v in P(TM) is called the set of limit directions of the vector field v. Rene Thom has conjectured that for each a [belongs to] M the set [q^-1] (a) [...] is semi-algebraic in P(T[sub a]M). In [3] Łojasiewicz proved that [...] is locally descibed by H[sub M]t[sub l],...,t[sub m] where H[sub M] is the sheaf of the real analytic functions on M. In (2), for any given open subset Omega of [R^n] we have associated an analytic subalgebra 0(Omega)[is a subset of] H(Omega) satisfying some conditions, and we have proved that the closure of any subset A [is a subset of] Omega which is described by global equations in 0(Omega) is a subset of the same type. In this paper we prove that if v : M -> TM is a vector field in some well defined subalgebra of the algebra of analytic functions on M, then the set of limit directions of v is also described by this algebra.

Borne d'invariant metrique pour une famille noetherienne

El Khadiri A.

Bulletin of the Polish Academy of Sciences. Mathematics

1998

Vol. 46, no 3

265-275

The motivation of this paper is a question asked by B. Teissier in [1]: if [fi] : M --> N is an analytic morphism between two real analytic manifolds M and N, and if K is a compact subanalytic set of M, then for every point x[sub 0] in [fi](K) there exists an open neighbourhood U of x[sub 0] in N and a constant [gamma] > 0 such that for all x in U and all (a, b) in the same connected component of [fi^-1(x) intersection of sets K], there exist a rectifiable curve in [fi-1(x) intersection of sets K joining a and b with length less than [gamma]. In this paper we prove the following statement: let [Omega] be an open set of [R^n], N a real analytic manifold, [fi] : [Omega] --> N a proper analytic morphism and [K is a subset of set Omega] an analytic subset of [Omega]. Then for every point y[sub 0] of N there an open neighbourhood U in N and a constant [eta] > 0 such that for all y in U there exists C[sub y] > 0 satisfying the following: for every points a, b of the same connected component of [fi^-1(y) intersection of sets K] there exists an analytic rectifiable curve [sigma] in [fi^-1(y) intersection of sets] K joining a and b with [absolute value of sigma is less than or equal to] C[sub y] [absolute value a - b^eta], where [absolute value of sigma] is the length of [sigma] and [absolute value a - b] is the euclidean distance between a and b.