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

# Ergodic

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

- 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 $N$
- 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)?

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

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

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?

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.

- Derive as a motivation for Axiom A flow mixingness
- 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

- Implications among
- unique ergodicity
- minimality
- entropy zero plus ergodicity

- Simple or semi-simple case
- Which one-parameter subgroups are unstable/stable foliations for some ergodic affine?
- Try a).

- Relate dynamical properties to representations of the group.
- K-property implies Bernoull?
- Weak mixing plus center s.s. implies Bernoulli? [
*For parts d and e, try nilmanifolds first*] - Ergodic implies there is a unique measure of maximal entropy?

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