On $\Sigma _A^+$, define $x \sim y $ if $\sigma ^n x = \sigma^m y $ for some $n,m $. Find topological invariants for $(\Sigma _A^+, \sim )$.

  1. Subshifts of finite type have good quotients with fixed points.
  2. Given a periodic point $p$  in $\Omega _i$, is there a Markov partition with $p$ in the interior of a rectangle?
  3. If $\partial^s \mathcal{C} \ne \emptyset$ does $\partial^s \mathcal{C}$ contain a periodic point?
  4. Do two subshifts of finite type with the same entropy have a common good quotient?
  5. $\Sigma _A, \Sigma _B$ 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  (one-sided or two-sided) 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. Entropy-conjugacy + 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

  1. speed of mixing
  2. asymptotic expression for the number of periodic orbits
  3. is $\varphi_1$ intrinsically ergodic?
  4. direct proof of mixing of measures
  5. analogue of $h(f) \geq \log |\lambda| $
  6. understand det$(Id - A) $ as an invariant; relation to $\zeta (0)$
  7. stability of $C$-density for attractors
  8. condition on $g$ so that $\Sigma_A (g) $ is analytically or $C^\infty $ embeddable as a basic set.
  9. 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 0-dimensional $\Omega _i$'s with the same entropy

Assume $\varphi _t $ $C$-dense. If $\nu $ is $\varphi _1 $ invariant is $\nu $ $\varphi $-invariant?

Symbolic dynamics for billiards

Structure of basic sets

  1. Classification via $(R,A)$
  2. Local Axiom A implies embeddable

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

  4. Phantom homology groups -shift equivalence of induced maps! 

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