Is there an expansive homeo of $\mathbb S^2$?

(Handel) If there exists a cross section for for all minimal sets of a flow, then there exist a global cross-section.

Which surfaces and which homotopy classes of homeos admit expansive homeos? distal homeos?

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

This problem isn't legible.

Geometric proof of unique ergodicity for irrational rotation of $\mathbb S^1$

Conjugacy between topology and measure theory

a. Weakest notion such that h(f) is an invariant

b. Entropy-conjugacy + equivalence on Baire sets; what are the equivalence relations on homeomorphisms or maps on $S^1$ and subshifts?

Does minimal or uniquely ergodic for a diffeo  $f$ implies $h(f) = 0$ (try homeo case too)?

  1. Is there a minimal diffeo homotopic to $\left( \begin{matrix} 2 & 1\\ 1& 1 \end{matrix} \right) \times Id. $ on $\mathbb T^3$?
  2. (Seifert conjecture) Minimal flow on $\mathbb S^3$.

Suppose $ F: C \to \mathbb{R}, C$ the Cantor set, has bounded total variation. Is there a homeo $g : [0,1] \to [0,1]$ and a diffeo (Lipschitz, maybe) $f:[0,1] \to \mathbb{R}$ such that \[ F = f\circ g |C.\]

Continuous systems in statistical mechanics. Is there a topological dynamics formulation?

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

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?

Topological Rokhlin's Theorem.