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 $f|N_\alpha$ 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)
References
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?
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 $(R,A)$
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Local Axiom A implies embeddable
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existence of canonical coordinates implies embeddable (compact abelian group actions ? are $\Omega $'s).
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Phantom homology groups -shift equivalence of induced maps!
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dim $\Omega$?; when is the quotient a manifold?