Lyapunov graphs and flows on surfaces
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- by K. A. de Rezende and R. D. Franzosa PDF
- Trans. Amer. Math. Soc. 340 (1993), 767-784 Request permission
Abstract:
In this paper, a characterization of Lyapunov graphs associated to smooth flows on surfaces is presented. We first obtain necessary and sufficient conditions for a Lyapunov graph to be associated to Morse-Smale flows and then generalize them to smooth flows. The methods employed in the proofs are of interest in their own right for they introduce the use of the Conley index in this context. Moreover, an algorithmic geometric construction of flows on surfaces is described.References
- Charles Conley, Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Mathematics, vol. 38, American Mathematical Society, Providence, R.I., 1978. MR 511133, DOI 10.1090/cbms/038
- Charles Conley and Eduard Zehnder, Morse-type index theory for flows and periodic solutions for Hamiltonian equations, Comm. Pure Appl. Math. 37 (1984), no. 2, 207–253. MR 733717, DOI 10.1002/cpa.3160370204
- John Franks, Nonsingular Smale flows on $S^3$, Topology 24 (1985), no. 3, 265–282. MR 815480, DOI 10.1016/0040-9383(85)90002-3
- John M. Franks, Homology and dynamical systems, CBMS Regional Conference Series in Mathematics, vol. 49, Published for the Conference Board of the Mathematical Sciences, Washington, D.C. by the American Mathematical Society, Providence, R.I., 1982. MR 669378, DOI 10.1090/cbms/049
- Frank Harary, Graph theory, Addison-Wesley Publishing Co., Reading, Mass.-Menlo Park, Calif.-London 1969. MR 0256911, DOI 10.21236/AD0705364 J. Milnor, Morse theory, Ann. of Math. Stud., no. 51, Princeton Univ. Press, Princeton, NJ, 1963. M. Peixoto, On the classification of flows on two-manifolds, Dynamical Systems, edited by M. M. Peixoto, Academic Press, New York, 1973, pp. 389-419.
- S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 228014, DOI 10.1090/S0002-9904-1967-11798-1
- F. Wesley Wilson Jr., Smoothing derivatives of functions and applications, Trans. Amer. Math. Soc. 139 (1969), 413–428. MR 251747, DOI 10.1090/S0002-9947-1969-0251747-9 C. Zeeman, Morse inequalities for diffeomorphisms with shoes and flows with solenoids, Dynamical Systems, Lecture Notes in Math., vol. 468, Springer, 1974, pp. 44-47.
Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 340 (1993), 767-784
- MSC: Primary 58F25; Secondary 54H20, 58E05
- DOI: https://doi.org/10.1090/S0002-9947-1993-1127155-9
- MathSciNet review: 1127155