Problem 7

(For part a.) Let $(\Omega, T)$ be an Axiom A basic set. Then there is a Markov partition given by rectangles $\{R_i\}_i$ and an adjacency matrix $A$ for the associated SFT $\Sigma_A$. Assume that the maximum diameter of the rectangles is at most 1/2 of an expansive constant for $T$. Let $\pi: \Sigma_A \rightarrow \Omega$ be the factor map: $\pi(x) = \cap_i T^{-i}(R_{x_i})$. Via $\pi$, $(\Omega, T)$ is topologically conjugate to $(Y,S)$ where $Y$ is the quotient of $\Sigma_A$ by the equivalence relation $x \sim x'$ iff for all $i$, $R_{x_i} \cap R_{x'_i} \ne \emptyset$ and $S$ is the map induced on the quotient by the shift. So, up to conjugacy, $(\Omega, T)$ is completely determined by the matrices $A$ and $R$ where $R_{ij} = 1$ if $R_i \cap R_j \ne \emptyset$ and 0 otherwise. The problem: classify Axiom A basic sets up to conjugacy by the matrices $A$ and $R$. This problem was mentioned in the syllabus for a seminar course that Rufus taught in Fall, 1973. Related material is contained in Fried [1] and Boyle-Buzzi [2].

References

In c. words in italics were added

In d. The ! is Rufus's. Can someone see the connection?

(For part d.) Isolated invariant sets $\Lambda$ have non-unique "filtration pairs", i.e. manifold neighborhoods $X$ with submanifold exit sets $E$ such that $\Lambda$ is the maximal invariant set in $X \setminus E$. The map induces a map on homology of the pair. The pairs are not unique and do not have unique homology, but the maps on homology are all shift equivalent. Rufus probably knew this and may have felt the shift equivalence class represented "phantom homology," i.e., some homology intrinsic to the isolated invariant set or basic set (see e.g. [1]) The concept of filtration pair was introduced in [2] where the intent was to study dynamical systems satisfying Axiom~A. This concept is related to the definition of  index pair introduced by C. Conley, [3], and studied by several authors (see [4], [5], and [6]). It is significantly less general than the definition of index pair found in [7]. All of these articles are interested in the homological invariants one can obtain associated to a filtration pair. D. Richeson and J. Franks in [8], show that shift equivalence holds at the topological level (not just at the homology level).

References

(For part d.) In [1] and later work, Putnam developed a homology theory for the class of Smale spaces (more general than Axiom A) that gives a Lefschetz-type formula for numbers of periodic points in terms of induced maps on homology.

References

(For part e.) The problem here is to decide, in terms of $A$ and $R$ as above, when the quotient space $\Omega$ is a manifold.

(For part b.) The question to decide if a combinatorics (R,A) can be realized by the basic set of an Axiom A diffeomorphism

of surface is discussed in:

C. Bonatti and R. Langevin, Difféomorphismes de Smale des surfaces. Astérisque 250 (1998).