Problem 65
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?
Comments
(For part b.) I don’t understand the indications.
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.
(For part c.) These questions make more sense for harmonic measures (cf. question 51 and [garnett1983foliations] )
(For part c. 2nd point). Can’t read. Plante’s theorem [plante1975foliations] here?
Maybe he is asking if polynomial growth is an invariant?
(For part e.) ( [plante1975foliations] ?)
The pointwise entropy of a foliation is defined in Definition 13.3, [hurder2009classifying].
The transverse Lyapunov exponents of foliations were defined in [hurder1988ergodic].