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Couples de fonctions et faux billards. (Couples of functions and wrong billiards). (French) Zbl 0691.58009
Let S be a closed $$C^{\infty}$$-differentiable curve in $$R^ 2$$ with a parametrization (f(t),g(t)) where f and g have pairwise disjoint critical values and no degenerated critical points. A set $$(x_ 1,...,x_{2q})$$ of disjoint points of S such that each pair $$(x_{2p+1},x_{2p+2})$$ lies on a vertical line and each pair $$(x_{2p},x_{2p+1})$$ and the pair $$(x_{2q},x_ 1)$$ lies on a horizontal line is called a cycle. A set $$(x_ 1,...,x_ n)$$ of points of S with the same properties as above except that of $$(x_ n,x_ 1)$$ and such that $$x_ 1$$ and $$x_ n$$ are the points at which S has a horizontal or vertical tangent is called a degenerated cycle. The author proves that if S intersects each horizontal and vertical line in at most two points, it has a cycle or a degenerated cycle.
The author investigates then couples of $$C^{\infty}$$-differentiable real functions f and g on a compact manifold V of dimension $$n\geq 2$$. He defines the notion of the cycle, of the transversal cusp, of the transversal fold and of stability for such a pair (f,g) and proves that the set of all (f,g) possessing transversal cusps or cycles is dense in $$C^{\infty}(V)\times C^{\infty}(V)$$ and that there is no stable pair (f,g) in $$C^{\infty}(V)$$.
Reviewer: J.Durdil

##### MSC:
 58C25 Differentiable maps on manifolds 58K99 Theory of singularities and catastrophe theory 57R45 Singularities of differentiable mappings in differential topology 57R50 Differential topological aspects of diffeomorphisms
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