By Ian F. Putnam

The writer develops a homology idea for Smale areas, which come with the fundamentals units for an Axiom A diffeomorphism. it's in response to parts. the 1st is a stronger model of Bowen's end result that each such procedure is a dead ringer for a shift of finite variety less than a finite-to-one issue map. the second one is Krieger's size crew invariant for shifts of finite variety. He proves a Lefschetz formulation which relates the variety of periodic issues of the process for a given interval to track facts from the motion of the dynamics at the homology teams. The life of this type of thought was once proposed through Bowen within the Seventies.

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**Additional resources for A Homology Theory for Smale Spaces (Memoirs of the American Mathematical Society)**

**Sample text**

Then π is s-bijective. The proof will be done in a series of Lemmas, beginning with the following quite easy one. 9. Let π : Y → X be a continuous map and let x0 be in X with π −1 {x0 } = {y1 , y2 , . . , yN } ﬁnite. For any > 0, there exists δ > 0 such that π −1 (X(x0 , δ)) ⊂ ∪N n=1 Y (yn , ). Proof. If there is no such δ, we may construct a sequence xk , k ≥ 1 in X converging to x0 and a sequence y k , k ≥ 1 with π(y k ) = xk and y k not in ∪N n=1 Y (yn , ). Passing to a convergent subsequence of the y k , let y be the limit point.

As < 0 , the closure of Wn is a compact subset of Y s (yn , 0 ). So the sequence y k has limit points; let y be one of them. By continuity, π(y) = x. On the other hand, there is a unique point y in Wn such that π(y ) = x. Thus, y and y are both in Y s (yn , 0 ) and have image x under π. As π is s-resolving, y = y and so y is in Wn . So the only limit point of the sequence y k is y and this completes the proof that π is a homeomorphism. Since Wn is an open subset of Y s (yn , ), we know that s Y (yn ) = ∪l≥0 ψ −l (Wn ) and the topology is the inductive limit topology.

By deﬁnition i ◦ t∗ = t(q)=p i(q), while t∗ ◦ i(p) = t(q)=i(p) q. We claim that i : {q | t(q) = p} → {q | t(q ) = i(p)} is a bijection. Since we suppose K ≥ 1, if q is such that t(q) = p, then t(i(q)) = i(t(q)) = i(p). Moreover, the map sending q 1 · · · q k to q 1 · · · q K pK is the inverse of i and this establishes the claim. The conclusion follows at once from this. The second part is proved in the same way and the last two statements are easy applications of the ﬁrst two. For a ﬁxed graph G, its higher block presentations, GK , K ≥ 1, all have the same Ds and Du invariants , stated precisely as follows.