# 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.