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

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

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?

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

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

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.

(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 $g$?

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

 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

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)


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.

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

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

Conditions on $M$ to admit Anosov $f$

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

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

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?

$C$-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?

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 $\mathbb S^3$.

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 $M$? 

Anosov diffeos


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

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?

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

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

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?