(Guillemin-Kazhdan) Invariant distributions for geodesic flows. Are they approximated by periodic orbit measures?
Zeta
- Subshifts of finite type have good quotients with fixed points.
- Given a periodic point $p$ in $\Omega _i$, is there a Markov partition with $p$ in the interior of a rectangle?
- If $\partial^s \mathcal{C} \ne \emptyset$ does $\partial^s \mathcal{C}$ contain a periodic point?
- Do two subshifts of finite type with the same entropy have a common good quotient?
- $\Sigma _A, \Sigma _B$ 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
- speed of mixing
- asymptotic expression for the number of periodic orbits
- is $\varphi_1$ intrinsically ergodic?
- direct proof of mixing of measures
- analogue of $h(f) \geq \log |\lambda| $
- understand det$(Id - A) $ as an invariant; relation to $\zeta (0)$
- stability of $C$-density for attractors
- condition on $g$ so that $\Sigma_A (g) $ is analytically or $C^\infty $ embeddable as a basic set.
- can a closed orbit of an Anosov flow be null homotopic?
Any `local' invariant (near fixed points) which are entropy-like.
Anosov diffeos
- Hypothesis on $H_1(M)$
- Fixed points
- $\Omega = M$
Kupka-Smale plus $h(f) >0 $ forces homoclinic points.
Zeta function for Axiom A flows and systems
- topological identification (try 1-dimensional $\Omega $ first); conjugacy invariance of $\zeta (0)$.
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For $C^\infty $ flows, $\zeta (s) $ has a meromorphic extension to the complex plane.
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Connection with Laplacian vs. geodesic results; automorphic forms.
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Anosov actions.