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 parts. the 1st is a much better model of Bowen's consequence that each such method is similar to a shift of finite kind lower than a finite-to-one issue map. the second one is Krieger's size workforce invariant for shifts of finite kind. He proves a Lefschetz formulation which relates the variety of periodic issues of the procedure for a given interval to track information from the motion of the dynamics at the homology teams. The life of any such conception was once proposed through Bowen within the Nineteen Seventies

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YL , z0 ) and has image (y0 , . . , yL ) under ρL, . Consider now the factor map ρL, : ΣL,0 (π) → YL (πs ). 3. That is, let M ≥ 0 and consider M +1 points in ΣL,0 (π) which all have the same image under ρL, . Using our earlier notation, this set is written as (ΣL,0 (π))M (ρL, ). Each element of such an M + 1-tuple has the form (y0 , . . , yL , z0 ) and the condition that they have the same image under ρL, simply means that the y0 , . . , yL entries of each one are all the same. 32 2. DYNAMICS Thus, we could list them as (y0 , .

2) For each M ≥ 0, let ρ,M : Σ0,M (π) → ZM (πu ) be the map defined by ρ,M (y0 , z0 , . . , zM ) = (z0 , . . , zM ). 11. Let π be an s/u-bijective pair for (X, ϕ). (1) For all L ≥ 0, ρL, is a u-bijective factor map. (2) For all M ≥ 0, ρ,M is an s-bijective factor map. Proof. We prove the first statement only. It is clear that ρL, is continuous and intertwines the dynamics. We check that it is onto. Let (y0 , . . , yL ) be in YL (πs ). As πu is onto, we may find z in Z such that πu (z) = πs (y0 ).

We conclude that pi = pj as desired. As we are discussing s/u-resolving maps between shifts of finite type, we describe a simple condition on the underlying graphs which is related. 16. Let G and H be graphs. A graph homomorphism θ : H → G is left-covering if it is surjective and, for every v in H 0 , the map θ : t−1 {v} → t−1 {θ(v)} is a bijection. Similarly, π is right-covering if it is surjective and, for every v in H 0 , the map θ : i−1 {v} → i−1 {θ(v)} is a bijection. The following result is obvious and we omit the proof.

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