# Homog

## Problem 157

On the closure $\overline T$ of Teichmüller space, consider a continuous parametrization $\overline T \times \Sigma_A^+ \to \mathbb S^2$ such that Image ($t, \Sigma_A^+ = \Lambda(\Gamma_t)$). Is the Hausdorff dimension of $\Lambda(\Gamma_t)$ continuous in $t \in \overline T$?

(Note, this problem was added by the editor to the end of Rufus' last paper [1])

### References

1. [bowen1979hausdorff] Bowen R.  1979.  Publications Mathématiques de l'IHÉS. 50:11–25.

## Problem 156

For a Kleinian group $\Gamma$, is the Hausdorff dimension of $\Lambda (\Gamma) <2$ if $\Lambda(\Gamma)$ is not the whole sphere?

(Note, this problem was added by the editor to the end of Rufus' last paper [1])

### References

1. [bowen1979hausdorff] Bowen R.  1979.  Publications Mathématiques de l'IHÉS. 50:11–25.

## Problem 143

Is $GR (\alpha)$ an algebraic integer for an automorphism $\alpha$ of solvable group?

## Problem 141

Same setting [as question 140]. Is the $GR$ (of $g$ on $\Gamma$) an algebraic number?

## Problem 140

Let $g : G/\Gamma \to G/\Gamma$ come from an automorphism $g$ of $G$. Is the entropy minimal in its homotopy class?  Is entropy a complete invariant for automorphisms of infranilmanifolds?

## 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 121

Calculate $h_\mu$ for $G/\Gamma$ finite measure, non compact.

## Problem 109

(Thom) Look at Markov partitions on $\mathbb{T}^n$ when all $\lambda_i$ are  distinct and real.

## Problem 107

Embed automorphisms of compact groups as basic sets.

## Problem 98

This problem is not clearly legible, but it is thought to say

Unique ergodicity in Lie groups. Case with finite area instead of compact homogeneous spaces

## Problem 87

If a translation by a group element on $G/\Gamma$ is minimal, is that element nilpotent in $\mathfrak G ?$ (i.e. has 0 entropy)

## Problem 62

Covering space for $\Sigma_A \to \mathbb{T}^2$ corresponding to $\mathbb{R}^2 \to \mathbb{T}^2$

## Problem 55

Entropy of automorphisms in algebra (groups, rings).

## Problem 13

In Parry's `Conjugate to linear' paper, what are the properties of the constructed measure? Does this work for equilibrium states too?

## Problem 40

Entropy of automorphisms of $C^\ast$-algebras

## Problem 35

Ergodic non hyperbolic automorphisms of $\mathbb{T}^n$. Are they quotients of subshifts of finite type? Do they satisfy specification?

## 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?