Abstract
We study germs of pairs of codimension one regular foliations in \({\mathbb{R}^3}\) . We show that the discriminant of the pair determines the topological type of the pair. We also consider various classifications of the singularities of the discriminant.
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Arnol’d, V.I., Guseĭ n-Zade, S.M., Varchenko, A.N.: Singularities of differentiable maps. Vol. I. The classification of critical points, caustics and wave fronts. Monographs in Mathematics, 82. Birkhuser (1985)
Bruce, J.W.: Classifications in singularity theory and their applications. New developments in singularity theory (Cambridge, 2000), 3–33, NATO Sci. Ser. II Math. Phys. Chem., 21, Kluwer Acad. Publ. (2001)
Bruce J.W.: On families of symmetric matrices. Mosc. Math. J. 3, 335–360 (2003)
Bruce J.W., Kirk N.P., du Plessis A.A.: Complete transversals and the classification of singularities. Nonlinearity 10, 253–275 (1997)
Bruce J.W., du Plessis A.A., Wall C.T.C.: Determinacy and unipotency. Invent. Math. 88, 521–554 (1987)
Bruce J.W., Tari F.: On families of square matrices. Proc. London Math. Soc. 89(3), 738–762 (2004)
Damon, J.N.: The unfolding and determinacy theorems for subgroups of \({{\mathcal A}}\) and \({{\mathcal K}}\) . Mem. Amer. Math. Soc. 50, no. 306 (1984)
Dufour J.-P.: Sur la stabilité des diagrams d’applications différentiables. Ann. Sci. École Norm. Sup. 10(4), 153–174 (1977)
Dufour J.-P.: Bi-stabilité des fronces. C. R. Acad. Sci. Paris Sér. A-B 285, A445–A448 (1977)
du Plessis, A.A., Wall, C.T.C.: The geometry of topological stability. London Mathematical Society Monographs. New Series, 9. Oxford Science Publications. Oxford University Press (1995)
Mancini S., Ruas M.A.S., Teixeira M.A.: On divergent diagrams of finite codimension. Port. Math. (N.S.) 59, 179–194 (2002)
Mather J.N.: Stability of C ∞ mappings, III: finitely determined map-germs. Publ. Math., IHES 35, 279–308 (1969)
Mather J.N.: Stability of C ∞ mappings, IV: classification of stable germs by \({{\mathcal R}}\) -algebras. Publ. Math., IHES 37, 223–248 (1969)
Oliveira R.D.S., Tari F.: Topological classification of pairs of regular foliations in the plane. Hokkaido Math. J. 31, 523–537 (2002)
Szlenk W.: An introduction to the theory of smooth dynamical systems. Wiley, PWIV- Polish Scientific Publishers, Warzawa (1984)
Tari, F.: Two-parameter families of binary differential equations. (to appear) Discrete Contin. Dynam. Syst.
Tari F.: Two-parameter families of implicit differential equations. Discrete Contin. Dynam. Syst. 13, 139–162 (2005)
Wall C.T.C.: Finite determinacy of smooth map-germs. Bull. London Math. Soc. 13, 481–539 (1981)
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Martins, L.F., Oliveira, R.D.S. & Tari, F. On pairs of regular foliations in \({\mathbb{R}^3}\) and singularities of map-germs. Geom Dedicata 135, 103–118 (2008). https://doi.org/10.1007/s10711-008-9265-x
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DOI: https://doi.org/10.1007/s10711-008-9265-x