Abstract
Of course, rough means structurally stable. Introduced by Andronov and Pontryagin, this property unifies foliations whose “phase portrait” is not affected by the small “irregularities” in the topological space F r(M). Fortunately, rough foliations are typical on compact orientable surfaces. If the surface is non orientable or non compact, then rough foliations are no longer typical.
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Bibliographic Notes
Andronov, A. A. and Pontryagin, L. S., 1937 Systèmes grossiers,Comptes Rendus de l’Académie des Sciences de l’URSS 14(5), 247–250.
Peixoto, M. M., 1962 Structural stability on two-dimensional manifolds, Topology 1, 101–120.
Bronstein, I. and Nikolaev I., 1998 Structurally stable fields of line elements on surfaces, Nonlinear Analysis 34, 461–477.
Gufnez, V., 1988 Positive quadratic differential forms and foliations with singularities on surfaces, Trans. Amer. Math. Soc. 309, 477–502.
Liousse, I., 1995 Dynamique générique des feuilletages transversalement affines des surfaces, Bull. Soc. math. France 123, 493–516.
Meyer, K. R., 1968 Energy functions for Morse-Smale systems, Amer. J. of Math. 90. 1031–1040.
Franks. J., 1979 Morse-Smale flows and homotopy theory, Topology 18, 199–215.
Peixoto, M. M., 1962 Structural stability on two-dimensional manifolds, Topology 1, 101–120.
Nikolaev, I., 1998 Graphs and flows on surfaces, Ergod. Th. and Dyn. Syst. 18, 207–220.
Bronstein, I. and Nikolaev I., 1997 Peixoto graphs of Morse-Smale foliations on surfaces, Topology and its Appl. 77, 19–36.
Bronstein, I. and Nikolaev I., 1994 On Peixoto graphs of Morse-Smale foliations,CRM-2214, Univ. de Montréal, 23p, Preprint.
Andronov, A. A. and Leontovich, E. A., 1965 Dynamical systems of the first order of stability in the plane,Mat. Sbornik 68, 328–372, MR 33# 2866
Sotomayor, J., 1968 Generic one-parameter families of vector fields on two-dimensional manifolds, Bull. Amer. Math. Soc. 74, 722–726.
Sotomayor, J., 1973 Generic one-parameter families of vector fields on two-dimensional manifolds, Publ. Math. IHES 43, 5–46.
Nikolaev, I., 1994 Foliations with singularities of semi-integer index,CRM-2206, Université de Montréal, 33p, Preprint.
Gutiérrez, C. and Sotomayor, J., 1990 Periodic lines of curvature from Darbouxian umbilical connections,Lecture Notes in Math. 1455, 196–229, Springer.
Gutiérrez, C. and Sotomayor, J., 1994 Lines of Curvature and Umbilical Points on Surfaces,Instituto de Matematica Pura e Aplicada, Rio de Janeiro, ISBN 85–244–0057–9.
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© 2001 Springer-Verlag Berlin Heidelberg
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Nikolaev, I. (2001). Morse-Smale Foliations. In: Foliations on Surfaces. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol 41. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04524-4_3
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DOI: https://doi.org/10.1007/978-3-662-04524-4_3
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