Is $GR (\alpha) $ an algebraic integer for an automorphism $\alpha$ of solvable group?
Topology
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 $fN_\alpha$ 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: \[ \frac {1}{1t} \; = \; \Pi _{n= 0}^\infty (1 +t^{2^n}) \] 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. 200202)
References
For Anosov flow $\varphi_t $ on $M$, try to approximate curves in $M$ by pseudoorbits 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?
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
 Ambrose Kakutani (in particular, question 39)
 Does mixing make any sense? (use category, differentiability, $C^\infty$, analytic structure)
 Averaging procedure difficulties:
 ergodic theorems, existence of invariant measures, ergodic decomposition, unique ergodicity and uniform convergence.
 polynomial growth ...
 Look at some specific foliations
 Plante's stuff on connections with homology
 Does pointwise entropy make sense?
$C$dense (mixing) Axiom A flows
 speed of mixing
 asymptotic expression for the number of periodic orbits
 is $\varphi_1$ intrinsically ergodic?
 direct proof of mixing of measures
 analogue of $h(f) \geq \log \lambda $
 understand det$(Id  A) $ as an invariant; relation to $\zeta (0)$
 stability of $C$density for attractors
 condition on $g$ so that $\Sigma_A (g) $ is analytically or $C^\infty $ embeddable as a basic set.
 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)?
 Is there a minimal diffeo homotopic to $\left( \begin{matrix} 2 & 1\\ 1& 1 \end{matrix} \right) \times Id. $ on $\mathbb T^3$?
 (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
 Hypothesis on $H_1(M)$
 Fixed points
 $\Omega = M$
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 $
 for diffeos
 $\Omega $ finite plus hyperbolic
 Axiom A with cycles.
Structure of basic sets
 Classification via $(R,A)$

Local Axiom A implies embeddable

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

Phantom homology groups shift equivalence of induced maps!

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