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

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 according to elements. the 1st is a stronger model of Bowen's end result that each such approach is clone of a shift of finite variety lower than a finite-to-one issue map. the second one is Krieger's measurement workforce invariant for shifts of finite sort. He proves a Lefschetz formulation which relates the variety of periodic issues of the method for a given interval to track facts from the motion of the dynamics at the homology teams. The lifestyles of this sort of thought used to be proposed through Bowen within the Nineteen Seventies

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The proof of the third part is analogous to that of the second. The fourth part follows from the second using l1 = l and l2 = α(l) and the third using m1 = m and m2 = β(m). For the ﬁfth part, we begin by choosing (y0 , . . , yL , z0 , . . , 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.

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.

First, since x0 is periodic, so is each yn . Choose p ≥ 1 such that ψ p (yn ) = yn , for all 1 ≤ n ≤ N , and hence ϕp (x0 ) = x0 . The system (Y, ψ p ) is also a Smale space. Choose 0 > > 0 to be less than the Smale space constant for this. Also, choose suﬃciently small so that the sets Y (yn , ), 1 ≤ n ≤ N are pairwise disjoint and so that ψ p (Y (yn , )) ∩ B(ym , ) = ∅, for m = n. 9 to ﬁnd δ such that π −1 {x} ⊂ ∪N n=1 Y (yn , ). Now suppose that x is in X s (x0 , δ) and π(y) = x. It follows that y is in Y (ym , ), for some m.