Read e-book online A homology theory for Smale spaces PDF

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 truly is according to elements. the 1st is a better model of Bowen's consequence that each such process is a dead ringer for a shift of finite sort less than 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 approach for a given interval to track facts from the motion of the dynamics at the homology teams. The life of one of these thought used to be proposed by means of Bowen within the Seventies

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If (X, ϕ) is a non-wandering Smale space, then there exists an s/u-bijective pair for (X, ϕ). Proof. 13, it suffices to consider the case that (X, ϕ) is irreducible. 5 of [15] to find an irreducible shift of finite type, (Σ, σ), an irreducible Smale space, (Y, ψ), and factor maps π1 : (Σ, σ) → (Y, ψ), π2 : (Y, ψ) → (X, ϕ) where π1 is u-resolving and π2 is s-resolving. 8. Let πs = π2 . 12, and hence totally disconnected. A similar argument shows the existence of (Z, ζ, πu ). 4. Let π = (Y, ψ, πs , Z, ζ, πu ) be an s/u-bijective pair for the Smale space (X, ϕ).

11. Let x be any point in B(x0 , δ). We know that s π −1 ({x}) is contained in ∪N n=1 Y (yn , 0 ). As the map π is s-resolving, it is injective when restricted to each of the sets Y s (yn , ). This means that π −1 {x} contains 26 2. DYNAMICS at most one point in each of these sets. On the other hand, it follows from our choice of x0 that π −1 {x} contains at least N points. We conclude that, for each n, π −1 {x} ∩ Y s (yn , ) contains exactly one point. Let Wn = π −1 (X(x, δ)) ∩ Y s (yn , ). The argument above shows that π is a bijection from Wn to X s (x0 , δ), for each n.

YL , z0 , . . , zM ) is in ΣL,M (π) and satisfies ek (yl1 , zm1 ) = ek (yl2 , zm2 ) for some 0 ≤ l1 , l2 ≤ L, some 0 ≤ m1 , m2 ≤ M and all 1 − K0 ≤ k ≤ K0 , then (1) ek (yl1 , zm ) = ek (yl2 , zm ), for all k ≤ 0 and 0 ≤ m ≤ M , and (2) ek (yl , zm1 ) = ek (yl , zm2 ), for all k ≥ 0 and 0 ≤ l ≤ L. Proof. We use the continuity of the map t ◦ e0 : Σ(π) → G0 to assert that there is a constant 0 < < Y , Z such that d(y, y ), d(z, z ) < implies t(e0 (y, z)) = t(e0 (y , z )). Next, we choose K0 ≥ 1 such that d(e(y, z), e(y , z )) ≤ 2−K0 implies d(y, y ), d(z, z ) < .

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A homology theory for Smale spaces by Ian F. Putnam

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