# Zeta

## Problem 147

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

## Problem 146

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 $\sigma_B |\Sigma_B$?

## Problem 131

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

## Problem 126

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

## Problem 110

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.

## Problem 104

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.....

## Problem 103

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

## Problem 102

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

## Problem 54

$C$-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?

## Problem 45

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

## Problem 31

Anosov diffeos

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

## Problem 28

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

## Problem 6

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.