On $\Sigma _A^+$, define $x \sim y $ if $\sigma ^n x = \sigma^m y $ for some $n,m $. Find topological invariants for $(\Sigma _A^+, \sim )$.
SymDyn
 Subshifts of finite type have good quotients with fixed points.
 Given a periodic point $p$ in $\Omega _i$, is there a Markov partition with $p$ in the interior of a rectangle?
 If $\partial^s \mathcal{C} \ne \emptyset$ does $\partial^s \mathcal{C}$ contain a periodic point?
 Do two subshifts of finite type with the same entropy have a common good quotient?
 $\Sigma _A, \Sigma _B$ aperiodic, $h (\sigma _A ) < h(\sigma_B). $ Does $\sigma_A \Sigma_A$ embed in $\sigma_B \Sigma_B$?
Are polynomial growth foliations hyperfinite? Is $W^{ws} $ on $\Sigma ^+_{\{0,1\}} $ Borel hyperfinite?
Horocycle and geodesic flows for $SL(2,R)/SL(2,Z)$:
 min u.e. almost
 something about symbolic dynamics and continued fractions?
For a subshift $\Sigma _A$, $A$ on stable torus or stable $\mathbb{R}^n$ is an invariant. Describe it invariantly.
($\mathbb{R}^n = ?$, $A = ?$)
$det (I  A) $ as a group invariant for the weak foliation $W^{wu}$, seen as a subgroup lying in some $H_m (M, \mathbb S^1)$?
(Thom) Look at Markov partitions on $\mathbb{T}^n$ when all $\lambda_i $ are distinct and real.
Let a group $G$ be given by a generator $S$ and relations. Consider the set $V$ of reduced (onesided or twosided) infinite words. What is $V$? Is it intrinsically ergodic? What is the entropy?
Is the multiplicity of $1$ as an eigenvalue of $A$ a flow conjugacy invariant of $\Sigma _A$? How about $\Pi_{\lambda_i \not = 1} (1 \lambda_i) $?
Markov partitions for algebraic geometry examples (Ruelle).
Conjugacy between topology and measure theory
a. Weakest notion such that h(f) is an invariant
b. Entropyconjugacy + equivalence on Baire sets; what are the equivalence relations on homeomorphisms or maps on $S^1$ and subshifts?
Covering space for $\Sigma_A \to \mathbb{T}^2$ corresponding to $\mathbb{R}^2 \to \mathbb{T}^2$
$C$dense (mixing) Axiom A flows
 speed of mixing
 asymptotic expression for the number of periodic orbits
 is $\varphi_1$ intrinsically ergodic?
 direct proof of mixing of measures
 analogue of $h(f) \geq \log \lambda $
 understand det$(Id  A) $ as an invariant; relation to $\zeta (0)$
 stability of $C$density for attractors
 condition on $g$ so that $\Sigma_A (g) $ is analytically or $C^\infty $ embeddable as a basic set.
 can a closed orbit of an Anosov flow be null homotopic?
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?
Ergodic non hyperbolic automorphisms of $\mathbb{T}^n$. Are they quotients of subshifts of finite type? Do they satisfy specification?
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 $g$.
Classify symbolic systems with specification.
Bifurcation of Axiom A in terms of symbols.
Canonical $C^0$ perturbation of Anosov diffeo to 0dimensional $\Omega _i$'s with the same entropy
Assume $\varphi _t $ $C$dense. If $\nu $ is $\varphi _1 $ invariant is $\nu $ $\varphi $invariant?
Structure of basic sets
 Classification via $(R,A)$

Local Axiom A implies embeddable

existence of canonical coordinates implies embeddable (compact abelian group actions ? are $\Omega $'s).

Phantom homology groups shift equivalence of induced maps!

dim $\Omega$?; when is the quotient a manifold?