# Topology

## Problem 143

Is $GR (\alpha)$ an algebraic integer for an automorphism $\alpha$ of solvable group?

## Problem 141

Same setting [as question 140]. Is the $GR$ (of $g$ on $\Gamma$) an algebraic number?

## Problem 140

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?

## Problem 130

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

## 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 122

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.

## Problem 117

(Sullivan) Show that $|\lambda| = 1$ for $f: M \to M$, where $f$ is  $C^1$ and distal and $\lambda$ is an eigenvalue of $f_\ast$. Is $f \sim g$ for some Morse Smale $g$?

## Problem 114

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

## 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 90

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 , pp. 200-202)

## Problem 89

For Anosov flow $\varphi_t$ on $M$, try to approximate curves in $M$ by pseudo-orbits and compute $\pi_1(M)$ . . . as in Morse theory.

## Problem 86

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

## Problem 76

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

## Problem 75

Conditions on $M$ to admit Anosov $f$

## Problem 73

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

## Problem 71

Cancellation of $\Omega_i$. Simplest $f$ in an iosotopy class.

## Problem 65

Foliation ergodic theory

1. Ambrose Kakutani (in particular, question 39)
2. Does mixing make any sense? (use category, differentiability, $C^\infty$, analytic structure)
3. Averaging procedure difficulties:
• ergodic theorems, existence of invariant measures, ergodic decomposition, unique ergodicity and uniform convergence.
• polynomial growth  ...
4. Look at some specific foliations
5. Plante's stuff on connections with homology
6. Does pointwise entropy make sense?

## 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 49

Does minimal or uniquely ergodic for a diffeo  $f$ implies $h(f) = 0$ (try homeo case too)?

1. Is there a minimal diffeo homotopic to $\left( \begin{matrix} 2 & 1\\ 1& 1 \end{matrix} \right) \times Id.$ on $\mathbb T^3$?
2. (Seifert conjecture) Minimal flow on $\mathbb S^3$.

## Problem 38

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 $M$?

## Problem 31

Anosov diffeos

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

## Problem 30

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?

## Problem 12

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

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

## Problem 7

Structure of basic sets

1. Classification via $(R,A)$
2. Local Axiom A implies embeddable

3. existence of canonical coordinates implies embeddable (compact abelian group actions ? are $\Omega$'s).

4. Phantom homology groups -shift equivalence of induced maps!

5. dim $\Omega$?; when is the quotient a manifold?