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4 results

  • Cohomology of chain recurrent sets

  • Morris W. Hirsch, Charles C. Pugh
  • Journal:
    Ergodic Theory and Dynamical Systems / Volume 8 / Issue 1 / March 1988
    Published online by Cambridge University Press:
    19 September 2008, pp. 73-80
    Print publication:
    March 1988
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  • Let ϕ be a flow on a compact metric space Λ and let p ∈ Λ be chain recurrent. We show that (Λ; ℝ) ≠ 0 if dimp Λ = 1 or if p belongs to a section of ϕ. Applications to planar flows and to smooth flows are given.

  • The C1,1 conclusions in Gromov's theory

  • Charles C. Pugh
  • Journal:
    Ergodic Theory and Dynamical Systems / Volume 7 / Issue 1 / March 1987
    Published online by Cambridge University Press:
    19 September 2008, pp. 133-147
    Print publication:
    March 1987
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  • According to M. Gromov, any sequence of Riemann manifolds with uniformly bounded geometry has a subsequence that converges to a limit. It is shown here that this limit Riemann structure is Lipschitz, generates a Lipschitz geodesic flow, and consequently, as Gromov asserted, the limit distance function is of class C1,1. Sharpness of the results is discussed. A simple, extrinsic proof of Gromov's Theorem is included.

  • The C1 Closing Lemma, including Hamiltonians

  • Charles C. Pugh, Clark Robinson
  • Journal:
    Ergodic Theory and Dynamical Systems / Volume 3 / Issue 2 / June 1983
    Published online by Cambridge University Press:
    19 September 2008, pp. 261-313
    Print publication:
    June 1983
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  • An Axiom of Lift for classes of dynamical systems is formulated. It is shown to imply the Closing Lemma. The Lift Axiom is then verified for dynamical systems ranging from C1 diffeomorphisms to C1 Hamiltonian vector fields.