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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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