A homology theory for Smale spaces by Ian F. Putnam

By Ian F. Putnam

The writer develops a homology idea for Smale areas, which come with the fundamentals units for an Axiom A diffeomorphism. it truly is in response to materials. the 1st is a much better model of Bowen's consequence that each such procedure is a dead ringer for a shift of finite kind less than a finite-to-one issue map. the second one is Krieger's measurement crew invariant for shifts of finite variety. 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 one of these idea used to be proposed by means of Bowen within the Nineteen Seventies

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A homology theory for Smale spaces

The writer develops a homology concept for Smale areas, which come with the fundamentals units for an Axiom A diffeomorphism. it really is in keeping with constituents. the 1st is a more robust model of Bowen's consequence that each such method is clone of 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 variety.

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Similarly, the edge set G1L,M is the set of those (L+1)×(M +1) arrays of entries of G1 obtained as e0 (yl , zm ), 0 ≤ l ≤ L, 0 ≤ m ≤ M , where (y0 , . . , yL , z0 , . . , zM ) is in ΣL,M (π). The definition of the maps i and t are obvious. 5, and we omit it. 9. Let π be an s/u-bijective pair for (X, ϕ) and suppose that G is a presentation of π. Then for every L, M ≥ 0, (ΣL,M (π), σ) ∼ = (ΣGL,M , σ). Although it is not needed now, it will be convenient for us to have other descriptions of these systems.

Let p be in GK . By definition i ◦ t∗ = t(q)=p i(q), while t∗ ◦ i(p) = t(q)=i(p) q. We claim that i : {q | t(q) = p} → {q | t(q ) = i(p)} is a bijection. Since we suppose K ≥ 1, if q is such that t(q) = p, then t(i(q)) = i(t(q)) = i(p). Moreover, the map sending q 1 · · · q k to q 1 · · · q K pK is the inverse of i and this establishes the claim. The conclusion follows at once from this. The second part is proved in the same way and the last two statements are easy applications of the first two. For a fixed graph G, its higher block presentations, GK , K ≥ 1, all have the same Ds and Du invariants , stated precisely as follows.

Let (y0 , . . , yL ) be in YL (πs ). As πu is onto, we may find z in Z such that πu (z) = πs (y0 ). Then (y0 , . . , yL , z) is in ΣL,0 (π) and its image under ρL, is (y0 , . . , yL ). Next, we check that ρL, is u-resolving. Suppose that (y0 , . . , yL , z0 ) and (y0 , . . , yL , z0 ) are unstably equivalent and have the same image under ρL, . The first fact implies, in particular, that z0 and z0 are unstably equivalent. The second fact just means that (y0 , . . , yL ) = (y0 , . . , yL ). Since the points are in ΣL,0 , we also have πu (z0 ) = πs (y0 ) = πs (y0 ) = πu (z0 ).

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