Problem 54

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$C$-dense (mixing) Axiom A flows

  1. speed of mixing
  2. asymptotic expression for the number of periodic orbits
  3. is $\varphi_1$ intrinsically ergodic?
  4. direct proof of mixing of measures
  5. analogue of $h(f) \geq \log |\lambda| $
  6. understand det$(Id - A) $ as an invariant; relation to $\zeta (0)$
  7. stability of $C$-density for attractors
  8. condition on $g$ so that $\Sigma_A (g) $ is analytically or $C^\infty $ embeddable as a basic set.
  9. can a closed orbit of an Anosov flow be null homotopic?


bonatti's picture

The answer to part 9 is known to be negative (no zero homotopic orbits) for codimension $1$ Anosov flow,

See Plante, J. F.; Thurston, W. P.
Anosov flows and the fundamental group.
Topology 11 (1972), 147–150.

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