Lyapunov graphs and flows on surfaces

Authors:
K. A. de Rezende and R. D. Franzosa

Journal:
Trans. Amer. Math. Soc. **340** (1993), 767-784

MSC:
Primary 58F25; Secondary 54H20, 58E05

MathSciNet review:
1127155

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[C]**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****[CZ]**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**, 10.1002/cpa.3160370204**[F1]**John Franks,*Nonsingular Smale flows on 𝑆³*, Topology**24**(1985), no. 3, 265–282. MR**815480**, 10.1016/0040-9383(85)90002-3**[F2]**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****[H]**Frank Harary,*Graph theory*, Addison-Wesley Publishing Co., Reading, Mass.-Menlo Park, Calif.-London, 1969. MR**0256911****[M]**J. Milnor,*Morse theory*, Ann. of Math. Stud., no. 51, Princeton Univ. Press, Princeton, NJ, 1963.**[P]**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.**[Sm]**S. Smale,*Differentiable dynamical systems*, Bull. Amer. Math. Soc.**73**(1967), 747–817. MR**0228014**, 10.1090/S0002-9904-1967-11798-1**[W]**F. Wesley Wilson Jr.,*Smoothing derivatives of functions and applications*, Trans. Amer. Math. Soc.**139**(1969), 413–428. MR**0251747**, 10.1090/S0002-9947-1969-0251747-9**[Z]**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.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
58F25,
54H20,
58E05

Retrieve articles in all journals with MSC: 58F25, 54H20, 58E05

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1993-1127155-9

Article copyright:
© Copyright 1993
American Mathematical Society