Local sections of flows on manifolds
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- by William C. Chewning and R. Steven Owen PDF
- Proc. Amer. Math. Soc. 49 (1975), 71-77 Request permission
Abstract:
We show the existence of local sections for dynamical systems on manifolds which are themselves generalized manifolds of codimension one. One implication of this is that the level surfaces of Lyapunov functions on three-manifolds are two-manifolds. This fact is used to study the geometry of stable attractors in three-manifolds.References
- N. P. Bhatia and G. P. Szegö, Stability theory of dynamical systems, Die Grundlehren der mathematischen Wissenschaften, Band 161, Springer-Verlag, New York-Berlin, 1970. MR 0289890
- Marston Morse, A reduction of the Schoenflies extension problem, Bull. Amer. Math. Soc. 66 (1960), 113–115. MR 117694, DOI 10.1090/S0002-9904-1960-10420-X
- Morton Brown, The monotone union of open $n$-cells is an open $n$-cell, Proc. Amer. Math. Soc. 12 (1961), 812–814. MR 126835, DOI 10.1090/S0002-9939-1961-0126835-6
- W. C. Chewning, A dynamical system on $E^{4}$ neither isomorphic nor equivalent to a differential system, Bull. Amer. Math. Soc. 80 (1974), 150–153. MR 326781, DOI 10.1090/S0002-9904-1974-13396-3 C. H. Edwards, A characterization of tame curves in the $3$-sphere, Notices Amer. Math. Soc. 7 (1960), 875. Abstract #573-32.
- Otomar Hájek, Dynamical systems in the plane, Academic Press, London-New York, 1968. MR 0240418
- William S. Massey, Algebraic topology: An introduction, Harcourt, Brace & World, Inc., New York, 1967. MR 0211390
- Frank Raymond, Separation and union theorems for generalized manifolds with boundary, Michigan Math. J. 7 (1960), 7–21. MR 120638
- T. Benny Rushing, Topological embeddings, Pure and Applied Mathematics, Vol. 52, Academic Press, New York-London, 1973. MR 0348752
- Hassler Whitney, Cross-sections of curves in $3$-space, Duke Math. J. 4 (1938), no. 1, 222–226. MR 1546045, DOI 10.1215/S0012-7094-38-00416-8
- Raymond Louis Wilder, Topology of Manifolds, American Mathematical Society Colloquium Publications, Vol. 32, American Mathematical Society, New York, N. Y., 1949. MR 0029491
Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 49 (1975), 71-77
- MSC: Primary 58F10; Secondary 34C35
- DOI: https://doi.org/10.1090/S0002-9939-1975-0362404-0
- MathSciNet review: 0362404