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Topological Rokhlin's Theorem.

Classification of singularities by the local properties of the gradient flow.

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.

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

Statistics plus dynamics of transformations of $[0,1]$ - 'non-linear' $\beta$-expansions like examples.

Shub's entropy conjecture: $h(f) \geq \log |\lambda| $

1.  for diffeos
2.  $\Omega $ finite plus hyperbolic
3.  Axiom A with cycles.

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

Brownian motion or diffusion given a flow

Interpret $-\log \lambda^u$ as a potential function  (Kolmogorov's   idea on surfaces of negative curvature)?

Continuous systems in statistical mechanics. Is there a topological dynamics formulation?

Canonical embedding of Axiom A $\Omega _i$

Bifurcation of Axiom A in terms of symbols.

Entropy in Hamiltonian case; for P.D.E.'s? Relation to O.D.E.'s?

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

Fixing compact manifold $M$, what are the possible behaviors of the geodesic flows for all Riemannian metrics? For instance, if $\pi _1(M) = 0 ,$ does some geodesic flow have entropy 0?

Classify symbolic systems with specification.

If $\mu $ is an equilibrium state for some continuous $g$ on $\Sigma _N^+$, is $h_\mu >0$?

If a geodesic flow is expansive, is it an Anosov flow?

Let $\mathcal M$ be the space of Riemannian metrics on $M$ with volume 1.  What is $\{ h(\varphi_1), \varphi {\textrm { geodesic flow of }}\; g: g \in \mathcal M\} $? What is the relation with topological invariants of $M$? 

Entropy of automorphisms of $C^\ast $-algebras

Is $h(\varphi_t  | E)$ differentiable in $E$ for Hamiltonian flows? Any relation to classical or quantum statistical mechanics? 

How big is the set of equilibrium states of a function $g$ in the set of invariant measures?

Equilibrium states for 1-dimensional quantum lattice systems without finite range

Suppose $ F: C \to \mathbb{R}, C$ the Cantor set, has bounded total variation. Is there a homeo $g : [0,1] \to [0,1]$ and a diffeo (Lipschitz, maybe) $f:[0,1] \to \mathbb{R}$ such that \[ F = f\circ g |C.\]

Is there a transitive/ergodic diffeomophism on $\mathbb S^2, \mathbb D^2$?

Define $\Omega $(foliation). Does $h >0 $ make sense?

$C$-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?

Is the horocycle flow an expansive flow?

Is Gutzwiller's example an Anosov flow?

Study flows $H = V(r) + \frac{1}{2} mv^2$ for various (?) continuous $V(r)$. Statistical mechanics literature (Hénon -?, Toda,...)

Covering space for $\Sigma_A \to \mathbb{T}^2$ corresponding to $\mathbb{R}^2 \to \mathbb{T}^2$

Entropy of group actions. There is no smooth $\mathbb{R}^n $- (or $\mathbb{Z}^n$-) action with positive entropy when $n >1$. Is this true for all Lie groups (or lattices) of dimension greater than 1? Try $N$ nilpotent.

Central Limit Theorem, other strong statistics near an attractor of a diffeo.

Electric circuits

• Analogue computer for finding Axiom A examples
• Is noise sometimes due to hyperbolicity in the dynamics?

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

This problem was crossed out.

If $f$ is Axiom A, is there an Axiom A $g$, $C^0$ near $f$ with dim$\Omega (g) = d$ and $h(f) = h(g) ?$

Does $h(f) $ have a minimum in isotopy class?

Calculate $h$ for O.D.E. systems on $\mathbb{R}^n$, e.g. linear equations first

Reddy examples of expansive maps. Related to Anosov diffeos. Are expansive diffeos likely to be Anosov?

(Plante) A codimension one minimal foliation has at most one invariant measure

Invariant or approximatively invariant finite dimensional subspaces for Navier-Stokes equation

Algebraic varieties. Weil conjecture, cohomology. Any entropy here? Any relation to homology eigenvalues?

Index 0 fixed point of diffeo is removable by small perturbation (Hirsch)

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

Among degree $n$ polynomial maps of $[0,1]$ to itself, are Axiom A open and dense. Do bad ones form a stratified set? ...

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

KAM Theorem using $\mu $ uniquely ergodic  flows on $M$ without assuming $M = T^n$

This problem is not clearly legible, but it is thought to say

Unique ergodicity in Lie groups. Case with finite area instead of compact homogeneous spaces

This problem isn't clearly legible, but it is thought to say: Symbolic dynamics for Abraham-Smale, Newhouse, Lorentz

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

 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 multiplicity of $1$ as an eigenvalue of $A$ a flow conjugacy invariant of $\Sigma _A$? How about $\Pi_{\lambda_i \not = 1} (1- \lambda_i) $?

Let a group $G$ be given by a generator $S$ and relations. Consider the set $V$ of reduced  (one-sided or two-sided) infinite words. What is $V$?  Is it intrinsically ergodic? What is the entropy?

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.

For a hyperbolic attractor $\Lambda$ of dimension $r$, does $W^s(x) \cap \Lambda $contain a disk of dimension $k := r- Dim W^u (x)?$

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

For a subshift $\Sigma _A$, $A$ on stable torus or stable $\mathbb{R}^n$ is an invariant. Describe it invariantly.

($\mathbb{R}^n = ?$, $A = ?$)

This problem was crossed out. 

Entropy 0 for differentiable action of $G = \mathbb{R}^n $ or $\mathbb{Z}^n (n \geq 2).$ For nilpotent $G$. General Lie group $G$?

Fibration Theorem for $\log |\lambda| $? If $M = \cup _\alpha N_\alpha, N_\alpha $ submanifolds, with $f(N_\alpha ) = N_\alpha $, is the spectral radius of $f_\ast $ on $M$ $\le$ the sup of the spectral radius of $f_\ast $ on $N_\alpha$'s?

If $f|N_\alpha$ is an isometry, does the spectral radius of $f_\ast $ on $M$ $= 1$? Are all distal diffeos built up this way, i.e. by extensions where homology works?

Note: There is another Problem 122 that was crossed out.

Weil conjecture for basic sets.

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

When are suspensions of $R_\alpha $ and $R_\beta$ under bounded functions isomorphic?

Which surfaces and which homotopy classes of homeos admit expansive homeos? distal homeos?

$\dot X = Q(X) $ on $\mathbb{R}^3$, where $Q$ is quadratic. Is there a condition on the coefficients which guarantees a homoclinic point (a complicated attractor)? E.g. like for Reynolds numbers?

1. If $\varphi _t$ is flow on a homogeneous space $G/ \Gamma$ with positive entropy, then there exists a compact $\varphi _t $ invariant section for the action of $N$
2. If the flow has entropy 0 and is ergodic, does this mean that there is  no $N$?

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.)

Billiards in right angle triangles. Find examples when it is ergodic.

Let $g : G/\Gamma \to G/\Gamma $ come from an automorphism $g$ of $G$. Is the entropy minimal in its homotopy class?  Is entropy a complete invariant for automorphisms of infranilmanifolds?

Can the  adding machine be an invariant set for a $C^2$ mapping of the 2-disk? of the $n$-disk?

Question 136 with expansive instead of hyperbolic.

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 $\sigma_B |\Sigma_B$?

$\Gamma$ a Kleinian group with limit set $\Lambda$. Specification when there are no parabolic nor elliptic elements.

Is there an expansive homeo of $\mathbb S^2$?

$T:M \to M \;  C^\infty $ Anosov. $f \in C^r $ and $f(x) = u(x) - u(Tx) .$   Does  $u \in C^r?$ ($r\geq 2$).

The geodesic flow on surfaces of higher genus always has positive measure-theoretic entropy.

For a Kleinian group $\Gamma$, is the Hausdorff dimension of $\Lambda (\Gamma) <2$ if $\Lambda(\Gamma)$ is not the whole sphere?

(Note, this problem was added by the editor to the end of Rufus' last paper [1])

References