(Guillemin-Kazhdan) Invariant distributions for geodesic flows. Are they approximated by periodic orbit measures?

  1. Subshifts of finite type have good quotients with fixed points.
  2. Given a periodic point $p$  in $\Omega _i$, is there a Markov partition with $p$ in the interior of a rectangle?
  3. If $\partial^s \mathcal{C} \ne \emptyset$ does $\partial^s \mathcal{C}$ contain a periodic point?
  4. Do two subshifts of finite type with the same entropy have a common good quotient?
  5. $\Sigma _A, \Sigma _B$ aperiodic, $h (\sigma _A ) < h(\sigma_B). $ Does $\sigma_A |\Sigma_A$ embed in $\sigma_B |\Sigma_B$?

$h(f) $ given by periodic points for generic $C^1$ map from $I$ to $I$? (or generic continuous map?)

Put orientation into the $\zeta$ function of flows. What should $\zeta (0)$ be? Does $\zeta (0)$ depend only on $H_\ast (M_0, M_{sing}) ?$.

Topological entropy of the Frobenius map of an algebraic variety $V$. Related to Dim $V$, to the log of the radius of convergence of the zeta function? Relations to zeta functions and to Weil conjectures.

 Note: \[ \frac {1}{1-t} \; = \; \Pi _{n= 0}^\infty (1 +t^{2^n}) \] in $Z[[ t]]$. Is there a $C^2$ map $\mathbb D^2 \to \mathbb D^2$ so that.....

How can you write \[ 1 +t  +t^2 \; = \; \Pi _{i=0 }^\infty  (1\pm t^{n_i}) \] in $Z[[ t]] ?$

Is the false zeta function of a basic set ....? Define false zeta function for a flow basic set

$C$-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?

Any `local' invariant (near fixed points) which are entropy-like.

Anosov diffeos


  1. Hypothesis on $H_1(M)$
  2. Fixed points
  3. $\Omega = M$

Kupka-Smale plus $h(f) >0 $ forces homoclinic points.

Zeta function for Axiom A flows and systems

  1.  topological identification (try 1-dimensional $\Omega $ first); conjugacy invariance of $\zeta (0)$.
  2. For $C^\infty $ flows, $\zeta (s) $ has a meromorphic extension to the complex plane.

  3. Connection with Laplacian vs. geodesic results; automorphic forms.

  4. Anosov actions.