Billiards in right angle triangles. Find examples when it is ergodic.

 Ergodic smooth representative of a Dehn twist. With smooth invariant measure of positive entropy. Of maximal entropy.

  1. If $\varphi _t$ is flow on a homogeneous space $G/ \Gamma$ with positive entropy, then there exists a compact $\varphi _t $ invariant section for the action of $N$
  2. If the flow has entropy 0 and is ergodic, does this mean that there is  no $N$?

$C^ \infty $ diffeo of the 2-disk preserving a smooth measure $\mu $ with $h_\mu >0$? An ergodic example?

When are suspensions of $R_\alpha $ and $R_\beta$ under bounded functions isomorphic?

Central Limit Theorem for $\beta$-transform $x \mapsto (\beta x)$.

KAM Theorem using $\mu $ uniquely ergodic  flows on $M$ without assuming $M = T^n$

Does every manifold $M^n, n\geq 3$ admit a smooth Bernoulli flow (Ruelle)?

This problem isn't legible.

(Plante) A codimension one minimal foliation has at most one invariant measure

Infinite measure space automorphisms

Central Limit Theorem, other strong statistics near an attractor of a diffeo.

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?

Rokhlin theorem for countable pseudo group actions. Ergodic Theorems and averaging procedures.

Look for invariant measures of some standard foliations.

Is there a transitive/ergodic diffeomophism on $\mathbb S^2, \mathbb D^2$?

Ambrose-Kakutani Theorem for $\mathbb{R}^n$ actions

Construction of a 2-dimensional Hamiltonian diffeo. with an ergodic set of positive measure.

Assume $\varphi _t $ $C$-dense. If $\nu $ is $\varphi _1 $ invariant is $\nu $ $\varphi $-invariant?

Can you construct some Banach space so that $h_\mu $ is an eigenvalue of some canonical operator?

Brownian motion or diffusion given a flow

Renewal theorems for dependent random variables.


  1.  Derive as a motivation for Axiom A flow mixingness
  2.  How fast is the mixing for Axiom A flows?

Statistics plus dynamics of transformations of $[0,1]$ - 'non-linear' $\beta$-expansions like examples.

Homogenous dynamics

  1. Implications among
    • unique ergodicity
    • minimality
    • entropy zero plus ergodicity
  2. Simple or semi-simple case
    • Which one-parameter subgroups are unstable/stable foliations for some ergodic affine?
    • Try a).
  3. Relate dynamical properties to representations of the group.
  4. K-property implies Bernoull?
  5. Weak mixing plus center s.s. implies Bernoulli? [For parts d and e, try nilmanifolds first]
  6. Ergodic implies there is a unique measure of maximal entropy?

`Geometric' proof that weak mixing implies mixing for a full set of $t$.

Topological Rokhlin's Theorem.