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?

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 [1])


References

  1. [garnett1983foliations] Garnett L.  1983.  Foliations, the ergodic theorem and Brownian motion. Journal of Functional Analysis. 51:285–311.

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


References

  1. [plante1975foliations] Plante JF.  1975.  Foliations with measure preserving holonomy. Annals of Mathematics. :327–361.

Maybe he is asking if polynomial growth is an invariant?

(For part e.) ([1]?)


References

  1. [plante1975foliations] Plante JF.  1975.  Foliations with measure preserving holonomy. Annals of Mathematics. :327–361.
Steve Hurder's picture

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

Steve Hurder's picture

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

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