Problem 65
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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?
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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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