By Ian F. Putnam

ISBN-10: 1470409097

ISBN-13: 9781470409098

The writer develops a homology conception for Smale areas, which come with the fundamentals units for an Axiom A diffeomorphism. it really is according to materials. the 1st is a more robust model of Bowen's consequence that each such process is just like a shift of finite sort below a finite-to-one issue map. the second one is Krieger's size workforce invariant for shifts of finite variety. He proves a Lefschetz formulation which relates the variety of periodic issues of the procedure for a given interval to track facts from the motion of the dynamics at the homology teams. The lifestyles of any such idea was once proposed via Bowen within the Seventies

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**Example text**

The periodic points y and [y, y ] are stably equivalent and hence must be equal. Similarly, the periodic points y and [y, y ] are unstably equivalent and hence must be equal. We conclude that y = y as desired. 9 for local stable sets. 11. Let π : (Y, ψ) → (X, ϕ) be a factor map between Smale spaces and suppose x0 in X is periodic and π −1 {x0 } = {y1 , y2 , . . , yN }. Given 0 > 0, there exist 0 > > 0 and δ > 0 such that s π −1 (X s (x0 , δ)) ⊂ ∪N n=1 Y (yn , ). Proof. First, since x0 is periodic, so is each yn .

We begin considering a pair of graphs and a graph homomorphism between them and the associated factor map between shifts of ﬁnite type. 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 .

The group Du (Σ, σ) is deﬁned to be the free abelian group on the ∼-equivalence classes of CO u (Σ, σ), modulo the subgroup generated by [E ∪ F ] − [E] − [F ], where E, F and E ∪ F are in CO u (Σ, σ) and E and F are disjoint. 1 of [22]). It asserts that the dimension group associated to a shift of ﬁnite type presented by a graph G is the same as that of the underlying graph. This provides a concrete method of computing the invariant. It will be useful for us to describe this isomorphism in terms of the invariant Ds (Gk ), for all k ≥ 1.

### A homology theory for Smale spaces by Ian F. Putnam

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