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 response to constituents. the 1st is a more robust model of Bowen's consequence that each such method is just like a shift of finite sort lower than a finite-to-one issue map. the second one is Krieger's size crew invariant for shifts of finite sort. He proves a Lefschetz formulation which relates the variety of periodic issues of the approach for a given interval to track facts from the motion of the dynamics at the homology teams. The lifestyles of this sort of idea was once proposed by means of Bowen within the Seventies

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

**Example text**

The set of equivalence class in ZG0 × N in the relation ∼s is denoted Ds (G). It is an abelian group with the operation [a, m]s + [b, m]s = [a + b, m]s , for all a, b in ZG0 and m in N. The group Du (G) is obtained in analogous way, using the relation ∼u . We remark that an important aspect in Krieger’s theory is that the group ZG0 has a natural order structure: the map γ s is positive and the inductive limit as actually taken in the category of ordered abelian groups. ) For our purposes here, we will ignore the order structure, but it should presumably feature in further developments.

It will be useful for us to describe this isomorphism in terms of the invariant Ds (Gk ), for all k ≥ 1. Recall in the formulas below, that for any e in ΣG and integer k, e[k+1,k] = i(ek+1 ) = t(ek ), by convention. 3. Let G be a graph, (ΣG , σ) be the associated shift of ﬁnite type and k ≥ 1. (1) The map sending [ΣsG (e, 2−j )], e ∈ ΣG , j ≥ k to [e[1−j,k−j−1] , j − k + 1] extends to an isomorphism from Ds (ΣG , σ) to Ds (Gk ). (2) The map sending [ΣuG (e, 2−j )], e ∈ ΣG , j ≥ k to [e[j−k+2,j] , j − k + 1] extends to an order isomorphism from Du (ΣG , σ) to Du (Gk ).

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 ).