By Ian F. Putnam

The writer develops a homology thought for Smale areas, which come with the fundamentals units for an Axiom A diffeomorphism. it truly is in response to constituents. the 1st is a higher model of Bowen's consequence that each such method is a twin of a shift of finite variety lower than a finite-to-one issue map. the second one is Krieger's measurement staff invariant for shifts of finite variety. He proves a Lefschetz formulation which relates the variety of periodic issues of the approach for a given interval to track info from the motion of the dynamics at the homology teams. The lifestyles of this type of thought used to be proposed by means of Bowen within the Seventies

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

**Example text**

If the graph homomorphism is left-covering, then our estimates are both simpler to state and easier to prove. But we remind the reader that the case of the a homomorphism δn : GN (π) → GN −1 (π) cannot be assumed to have this property. 1. Let G, H be graphs and θ : H → G be a graph homomorphism. (1) If the associated map on the shift spaces is an s-resolving factor map, then there is a constant Kθ ≥ 0 such that, if e, f are in ΣH and are stably equivalent and k0 is an integer such that θ(e)k = θ(f )k , for all k ≥ k0 , then ek = f k , for all k ≥ k0 + Kθ .

Suppose that each point of (Y, ψ) is non-wandering. 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 , ).

ZM ) in GK L,M such that e[−K0 ,K−K0 ] (yl , zm ) = pl,m , for all l, m. 6 with l1 = l, l2 = α(l) and any m1 = m2 = m. It follows that ek (yl , zm ) = ek (yα(l) , zm ) with k ≤ 0. Applying k = −K yields the result. The proof of the last statement is analogous to the ﬁfth and we omit it. CHAPTER 3 Dimension groups In this chapter, we present background material on Krieger’s theory of dimension group invariants for shifts of ﬁnite type. The ﬁrst section presents some very simple observations on free abelian groups.