Ergodic

Problem 138

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

Problem 137

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

Problem 134

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 $N$?

Problem 133

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

Problem 128

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

Problem 111

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

Problem 96

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

Problem 94

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

Problem 93

This problem isn't legible.

Problem 82

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

Problem 79

Infinite measure space automorphisms

Problem 66

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

Problem 65

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?

Problem 61

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

Problem 51

Look for invariant measures of some standard foliations.

Problem 50

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

Problem 39

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

Problem 27

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

Problem 21

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

Problem 19

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

Problem 16

Brownian motion or diffusion given a flow

Problem 15

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?

Problem 10

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

Problem 5

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?

Problem 3

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

Problem 2

Topological Rokhlin's Theorem.