Francois Laudenbach

A Note on the Chas-Sullivan product

We give a finite dimensional approach to the Chas-Sullivan product on the free loop space of a manifold, orientable or not.

Dynamically ordered energy function for Morse-Smale diffeomorphisms on 3-manifolds

This paper deals with arbitrary Morse-Smale diffeomorphisms in dimension 3 and extends ideas from the authors’ previous studies where the gradient-like case was considered. We introduce a kind of Morse-Lyapunov function, called dynamically ordered, which fits well the dynamics of a diffeomorphism. The paper is devoted to finding conditions for the existence of such an energy function, that is, a function whose set of critical points coincides with the nonwandering set of the considered diffeomorphism. We show that necessary and sufficient conditions for the existence of a dynamically ordered energy function reduce to the type of the embedding of one-dimensional attractors and repellers, each of which is a union of zeroand one-dimensional unstable (stable) manifolds of periodic orbits of a given Morse-Smale diffeomorphism on a closed 3-manifold.

Quasi-energy function for diffeomorphisms with wild separatrices

AbstractWe consider the class G4 of Morse—Smale diffeomorphisms on

A Morse complex on manifolds with boundary

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A proof of Reidemeister-Singer’s theorem by Cerf’s methods

3 with nonwandering set consisting of four fixed points (namely, one saddle, two sinks, and one source). According to Pixton, this class contains a diffeomorphism that does not have an energy function, i.e., a Lyapunov function whose set of critical points coincides with the set of periodic points of the diffeomorphism itself. We define a quasi-energy function for any Morse—Smale diffeomorphism as a Lyapunov function with the least number of critical points. Next, we single out the class G4,1 ⊂ G4 of diffeomorphisms inducing a special Heegaard splitting of genus 1 of the sphere

On the flux of pseudo-Anosov homeomorphisms

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