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?