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2 results

  • Periodic Orbits for Generalized Gradient Flows

  • Sol Schwartzman
  • Journal:
    Canadian Mathematical Bulletin / Volume 38 / Issue 1 / 01 March 1995
    Published online by Cambridge University Press:
    20 November 2018, pp. 117-119
    Print publication:
    01 March 1995
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  • Let Mn be an n-dimensional compact oriented connected Riemannean manifold. It is proved that either of the following conditions is sufficient to insure that the flow defined by a generalized gradient vector field in Mn has either a stationary point or a periodic orbit:

    • a)Mn is the product of a circle with an (n — 1 ) dimensional manifold of non-zero Euler characteristic.

    • b)The (n — 1) dimensional Stiefel-Whitney class of Mn is different from zero and in addition Mn possesses no one-dimensional 2-torsion.

  • Topological Transitivity on the Torus

  • Sol Schwartzman
  • Journal:
    Canadian Mathematical Bulletin / Volume 37 / Issue 4 / 01 December 1994
    Published online by Cambridge University Press:
    20 November 2018, pp. 549-551
    Print publication:
    01 December 1994
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  • T. Ding has shown that a topologically transitive flow on the torus given by a real analytic vector field is orbitally equivalent to a Kronecker flow on the torus, modified so as to have a finite number of fixed points, provided the original flow had only a finite number of fixed points. In this paper it is shown that the assumption that there are only finitely many fixed points is unnecessary.