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Smoothing continuous flows on two-manifolds and recurrences

Published online by Cambridge University Press:  19 September 2008

Carlos Gutierrez
Carlos Gutierrez
Affiliation:
Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, 22.460-Rio de Janeiro-RJ-Brasil
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Abstract

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Let φ: ℝ × M → M be a continuous flow on a compact C∞ two-manifold M. It is proved that there exists a C1 flow ψ on M which is topologically equivalent to φ, and that the following conditions are equivalent:

(a) any minimal set of φ is trivial;

(b) φ is topologically equivalent to a C2 flow;

(c) φ is topologically equivalent to a C∞ flow.

Also proved is a structure and an existence theorem for continuous flows with non-trivial recurrence.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 6 , Issue 1 , March 1986 , pp. 17 - 44
Copyright
Copyright © Cambridge University Press 1986

References

REFERENCES

[B—S]Bhatia, N.P. & Szego, B. P.. Stability Theory of Dynamical Systems. Springer-Verlag: New York, (1970).CrossRefGoogle Scholar
[Chr]Cherry, T. M.. Analytic quasi-periodic curves of discontinuous type on a torus. Proc. Lond. Math. Soc. 44 (1938) 175–215.CrossRefGoogle Scholar
[Chw]Chewing, W.C.. A dynamical system on E4 neither isomorphic nor equivalent to a differential system. Bull. Amer. Math. Soc. 80 (1974), 150–154.CrossRefGoogle Scholar
[De]Denjoy, A.. Sur les courbes définies par les équations differentielles à la surface du tore. J. Mathématique 9 (11) (1932) 333–375.Google Scholar
[Ga]Gardiner, C. J.. The structure of flows exhibiting non-trivial recurrence on two-dimensional manifolds. J. Diff. Eqn. 57 (1985), 138–158.CrossRefGoogle Scholar
[Gu.1]Gutierrez, C.. Smooth non-orientable non-trivial recurrence on two-manifolds. J. Diff. Eq. 29 (3) (1978) 388–395.CrossRefGoogle Scholar
[Gu.2]Gutierrez, C.. Smoothability of Cherry flows on two-manifolds. In Springer Lecture Notes in Mathematics, 1007, Geometric Dynamics, Proc. Rio de Janeiro, pp. 308–331 (1981).CrossRefGoogle Scholar
[Gu.3]Gutierrez, C.. Smoothing foliations on two-manifolds. To appear.Google Scholar
[Gu.4]Gutierrez, C.. Structural Stability for flows on the torus with a cross-cap. Trans. Amer. Math.Soc. 241 (1978), 311–320.CrossRefGoogle Scholar
[Gu.5]Gutierrez, C.. Smoothing continuous flows and the converse of Denjoy-Schwartz Theorem. An. Acad. Brasil Cieênc. 51 no. 4 (1979), 581–589.Google Scholar
[Ha]Halmos, P. R.. Measure Theory. Van Nostrand (1950).CrossRefGoogle Scholar
[Hr]Harrison, J.. Unsmoothable diffeomorphisms. Ann. of Math. 102 (1975), 85–94.CrossRefGoogle Scholar
[Ht]Hartman, P.. Ordinary Differential Equations. John Wiley and Sons. Inc. (1964).Google Scholar
[Ka]Katok, A.. Interval exchange transformations and some special flows are not mixing. Israel J. Math. 35, (4) (1980), 301–310.CrossRefGoogle Scholar
[Ke]Keane, M.. Interval exchange transformations. Math. Z. 141 (1975), 25–31.CrossRefGoogle Scholar
[Le.1]Levitt, G.. Feuilletages des surfaces. Ann. Inst. Fourier 32 (2) (1982).CrossRefGoogle Scholar
[Le.2]Levitt, G.. La décomposition dynamique et la différentiability des feuilletages des surfaces. Preprint.Google Scholar
[Le.3]Levitt, G.. Pantalons et feuilletages des surfaces. Topology 21 (1) (1982) 9–33.CrossRefGoogle Scholar
[Ma]Markley, N.. The Poincaré-Bendixson theorem for the Klein bottle. Trans. Amer. Math. Soc. 135 (1969), 159–165.Google Scholar
[Me]Mendes, L. H.. Interval exchange transformations and foliations. Preprint: to appear in Bol. Soc. Bras. Mat.Google Scholar
[Mu]Munkres, J.. Obstructions to smoothing piecewise differentiable homeomorphisms. Ann. Math. 72 (1960) 521–554.CrossRefGoogle Scholar
[Ne.l]Neumann, D.. Central sequences in flows on 2-manifolds of finite genus. Proc. Amer. Math. Soc. 61 (1) (1976), 39–43.CrossRefGoogle Scholar
[Ne.2]Neumann, D.. Smoothing continuous flows on 2-manifolds. J. Diff. Eq. 28 (3) (1978), 327–344.CrossRefGoogle Scholar
[Pe]Peixoto, M.. Structural stability on two-dimensional manifolds. Topology 1 (1962) 101–120.CrossRefGoogle Scholar
[Ro]Rosenberg, H.. Labryinths in the disc and surfaces. Ann. of Math. 117 (1983), 1–33.CrossRefGoogle Scholar
[Sch]Schwartz, A. J.. A generalization of the Poincaré-Bendixson theorem to closed two-dimensional manifolds. Amer. J. Math. 85 (1963), 453–548.CrossRefGoogle Scholar
[Sm]Smale, S.. Differentiable dynamical systems. Bull. Amer. Math. Soc. 73 (1967), 747–817.CrossRefGoogle Scholar
[St]Stredder, P.. Morse foliations. Thesis, Warwick, 1976.Google Scholar
[Swe]Schweitzer, P. A.. Counterexamples to the Seifert conjecture and opening closed leaves of foliations. Ann. of Math. 100 (1974), 386–400.CrossRefGoogle Scholar
[S—T]Schwartz, A. J. & Thomas, E. S.. The depth of the center of 2-manifolds. Global Anal. Proc. Symp. Pure Math., 14 Amer. Math. Soc: Providence, R.I., 253–264 (1970).CrossRefGoogle Scholar
[Ve.l]Veech, W. A.. Interval exchange transformations. J. d' Analyse Math. 33 (1978), 222–272.CrossRefGoogle Scholar
[Ve.2]Veech, W. A.. Quasiminimal invariants for foliations of orientable closed surfaces. Preprint, Rice University.Google Scholar
[Wt]Whitney, H.. Regular families of curves. Ann. of Math. 34 (2) (1933), 244–270.CrossRefGoogle Scholar