Hyperbolic

$\varphi _t: T^1M \to T^1M$ Anosov geodesic flow and $V: M \to \mathbb{R}$ such that $\int V(\pi \varphi_t x ) = 0$ on every closed geodesic. Is $V$ identically $0$?

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 $\sigma_B |\Sigma_B$?

For most $C^2$ maps $f : [0,1] \to [0,1]$, for all $\epsilon >0$ there is a hyperbolic set $\Lambda$ such that $h(f|\Lambda) \geq h(f) - \epsilon.$ (See question 8c.)

Weil conjecture for basic sets.

(Thom) $Grad F$ for real analytic $F:\mathbb{R}^n \to \mathbb{R}$. Stratification of orbits near a singular point.

$det (I - A)$ as a group invariant for the weak foliation $W^{wu}$, seen as a subgroup lying in some $H_m (M, \mathbb S^1)$?

Central Limit Theorem for $\beta$-transform $x \mapsto (\beta x)$.

 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.....

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

(Doug (Lind?)) Find open partitions in $\Sigma _{1/2,1/2}$ that are not weakly Bernoulli. Find invariants of finitary codes.

Does every manifold $M^n, n\geq 3$ admit a smooth Bernoulli flow (Ruelle)?

If $f$ is Anosov on $M$ and $\tilde M$ contractible, what does $H^k(M)(\sim H^k (\pi _1(M)) )$ tell you via $f_\ast$ eigenvalue information? (See [1], pp. 200-202)

References

Unstable foliations of Anosov diffeos are given by some nilpotent group action.

Conditions on $M$ to admit Anosov $f$

If $f$ is Anosov and $g \sim f$, does $h(g) \geq h(f) ?$

Classify all Anosov systems or attractors (which $\Omega_i$ can occur as attractors?)

Computer programs for Axiom A attractor.

$\ell (f^n \gamma)$ grows slowly with $n$ for many curves $\gamma$ and Axiom A diffeos $f$.

Is $\varphi_1$ a continuity point for the entropy as a function of diffeos when $\varphi_t$ is Axiom A flow? an Anosov flow?

Anosov diffeos

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

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

Canonical $C^0$ perturbation of Anosov diffeo to 0-dimensional $\Omega _i$'s with the same entropy

Assume $\varphi _t$ $C$-dense. If $\nu$ is $\varphi _1$ invariant is $\nu$ $\varphi$-invariant?

Symbolic dynamics for billiards

Suspensions of diffeos. -  Are they generically not  (conjugated to) constant time suspensions? what is the strongest statement for Axiom A attractors?

Nonalgebraic Anosov diffeos. Classify 3-dimensional Anosov flows; Is the variable curvature surface geodesic flow conjugate to constant curvature?

Non Axiom A examples. Newhouse, Abraham-Smale, Simon, Lorenz, billiards.

1. Axiomatic description
2. Statistical properties
3. For all $\epsilon$, there exists a horseshoe $X_\epsilon$ inside with $h(f|X_\epsilon) \geq h(f) - \epsilon$
4. Statistical properties of Lorenz in particular
5. Any specification type property

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.