Are polynomial growth foliations hyperfinite? Is Wws on Σ+{0,1} Borel hyperfinite?
Foliation
det(I−A) as a group invariant for the weak foliation Wwu, seen as a subgroup lying in some Hm(M,S1)?
For a hyperbolic attractor Λ of dimension r, does Ws(x)∩Λcontain a disk of dimension k:=r−DimWu(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∞, 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?
Look for invariant measures of some standard foliations.