Are polynomial growth foliations hyperfinite? Is $W^{ws} $ on $\Sigma ^+_{\{0,1\}} $ Borel hyperfinite?

# Foliation

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

Unstable foliations of Anosov diffeos are given by some nilpotent group action.

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

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?

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

Look for invariant measures of some standard foliations.