If a $C^1$ Anosov preserves a smooth measure, is it an equilibrium state for $ - \log \lambda ^u ?$

# Thermo

codon frequencies via equilibrium states for ``some potential''?

Study flows $H = V(r) + \frac{1}{2} mv^2$ for various (?) continuous $V(r)$. Statistical mechanics literature (Hénon -?, Toda,...)

Equilibrium states for 1-dimensional quantum lattice systems without finite range

How big is the set of equilibrium states of a function $g$ in the set of invariant measures?

Definition of Gibbs measures for homeomorphisms? Relation to equilibrium states?

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

If $\mu $ is an equilibrium state for some continuous $g$ on $\Sigma _N^+$, is $h_\mu >0$?

Define $P(g)$, equilibrium states for certain noncontinuous $g$.

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

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?