# Problem 65

## Primary tabs

Foliation ergodic theory

1. Ambrose Kakutani (in particular, question 39)
2. Does mixing make any sense? (use category, differentiability, $C^\infty$, analytic structure)
3. Averaging procedure difficulties:
• ergodic theorems, existence of invariant measures, ergodic decomposition, unique ergodicity and uniform convergence.
• polynomial growth  ...
4. Look at some specific foliations
5. Plante's stuff on connections with homology
6. Does pointwise entropy make sense?

## Tags

### 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

1. [garnett1983foliations] Garnett L.  1983.  Journal of Functional Analysis. 51:285–311.

### FL

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

### References

1. [plante1975foliations] Plante JF.  1975.  Annals of Mathematics. :327–361.

### BHM

Maybe he is asking if polynomial growth is an invariant?

### FL

(For part e.) ([1]?)

### References

1. [plante1975foliations] Plante JF.  1975.  Annals of Mathematics. :327–361.

### 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].