We show some generic (robust) properties of smooth surfaces immersed in the real 3-space (Euclidean, affine or projective), in the neighbourhood of a godron (called also cusp of Gauss): an isolated parabolic point at which the (unique) asymptotic direction is tangent to the parabolic curve. With the help of these properties and a projective invariant that we associate to each godron we present all possible local configurations of the flecnodal curve at a generic swallowtail in R 3 . We present some global results, for instance: In a hyperbolic disc of a generic smooth surface, the flecnodal curve has an odd number of transverse self-intersections (hence at least one self-intersection).

Abstract. The focal curve of an immersed smooth curve γ : θ → γ(θ), in Euclidean space R m+1 , consists of the centres of its osculating hyperspheres. This curve may be parametrised in terms of the Frenet frame of γ (t, n1, . . . , nm), as Cγ (θ) = (γ+c1n1+c2n2+· · ·+cmnm)(θ), where the coefficients c1, . . . , cm−1 are smooth functions that we call the focal curvatures of γ. We discovered a remarkable formula relating the Euclidean curvatures κi, i = 1, . . . , m, of γ with its focal curvatures. We show that the focal curvatures satisfy a system of Frenet equations (not vectorial, but scalar!). We use the properties of the focal curvatures in order to give, for l = 1, . . . , m, necessary and sufficient conditions for the radius of the osculating l-dimensional sphere to be critical. We also give necessary and sufficient conditions for a point of γ to be a vertex. Finally, we show explicitly the relations of the Frenet frame and the Euclidean curvatures of γ with the Frenet frame and the Euclidean curvatures of its focal curve Cγ. (2000) : 51L15, 53A04, 53D12. Mathematics Subject Classification

No abstract

No abstract

We study and develop a very new object introduced by V.I. Arnold: a monad is a triple consisting of a finite set, a map from that finite set to itself and the monad graph which is the directed graph whose vertices are the elements of the finite set and whose arrows lead each vertex to its image (by the map). We consider the case in which the finite set entering in the monad definition is a finite group G and the map f : G → G is the Frobenius map f k : x → x k , for some k ∈ Z. We study the Frobenius dynamical system defined by the iteration of the monad f k , and also study the combinatorics and topology (i.e., the discrete invariants) of the monad graph. Our study provides useful information about several structures on the group associated to the monad graph. So, for example, several properties of the quadratic residues of finite commutative groups can be obtained in terms of the graph of the Frobenius monad f 2 : x → x 2 .