By Ian F. Putnam

The writer develops a homology conception for Smale areas, which come with the fundamentals units for an Axiom A diffeomorphism. it's in keeping with parts. the 1st is a far better model of Bowen's end result that each such process is similar to a shift of finite style lower than a finite-to-one issue map. the second one is Krieger's size team invariant for shifts of finite sort. He proves a Lefschetz formulation which relates the variety of periodic issues of the process for a given interval to track information from the motion of the dynamics at the homology teams. The lifestyles of this type of concept used to be proposed by way of Bowen within the Nineteen Seventies

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**Additional info for A homology theory for Smale spaces**

**Example text**

Let (X, ϕ) and (Y, ψ) be Smale spaces and let π : (Y, ψ) → (X, ϕ) be a map. We say that π is s-bijective (or u-bijective) if, for any y in Y , its restriction to Y s (y) (or Y u (y), respectively) is a bijection to X s (π(y)) (or X u (π(y)), respectively). It is relatively easy to ﬁnd an example of a map which is s-resolving, but not s-bijective and we will give one in a moment. However, one important distinction between the two cases should be pointed out at once. The image of a Smale space under an s-resolving map is not necessarily a Smale space.

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. Then y is k k not in ∪N n=1 Y (yn , ), since that set is open, while π(y) = limk π(y ) = limk x = x0 .

Similarly, X s (x0 ) = ∪l≥0 ϕ−l (X s (x0 , δ)) and the topology is the inductive limit topology. It follows at once that π is a homeomorphism from the former to the latter. Now we turn to arbitrary point y in Y and x = π(y) in X and show that π : Y s (y) → X s (x) is onto. We choose x0 and {y1 , . . , yN } to be periodic points as above so that π : Y s (yn ) → X s (x0 ) are homeomorphisms. By replacing x0 by another point in its orbit (which will satisfy the same condition), we may assume that x is in the closure of X s (x0 ).