# Thermo

## Problem 101

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

## Problem 85

codon frequencies via equilibrium states for some potential''?

## Problem 60

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

## Problem 46

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

## Problem 44

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

## Problem 43

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

## 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 34

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

## Problem 33

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

## Problem 20

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

## 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?