# Problem 65

## Primary tabs

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?

## Tags

## Comments

## FL

(For part b.) I don't understand the indications.

## BHM

The indications are diffeos, $C^\infty$, analytic and may be a reference to smoothness assumptions on foliations; the problem here is that even for 1D flows, mixing is not invariant under reparametrization.

## FL

(For part c.) These questions make more sense for harmonic measures (cf. question 51 and [1])

## References

## FL

(For part c. 2nd point). Can't read. Plante's theorem [1] here?

## References

## BHM

Maybe he is asking if polynomial growth is an invariant?

## FL

(For part e.) ([1]?)

## References

## The pointwise entropy of a

The pointwise entropy of a foliation is defined in Definition 13.3, [hurder2009classifying].

## The transverse Lyapunov

The transverse Lyapunov exponents of foliations were defined in [hurder1988ergodic].

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