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 idea for Smale areas, which come with the fundamentals units for an Axiom A diffeomorphism. it truly is in accordance with elements. the 1st is a far better model of Bowen's end result that each such process is a dead ringer for a shift of finite variety lower 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 this type of concept was once proposed through Bowen within the Nineteen Seventies

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Extra resources for A homology theory for Smale spaces

Example text

As we see from the proof above, there is a close relation between the constant Kρ of a factor map ρ : (ΣG , σ) → (X, ϕ) and that of the factor map δ0 : (Σ1 (ρ), σ) → (ΣG , σ). In fact, the same relation exists for all higher self-products of (ΣG , σ), as follows. The proof is straightforward and we omit it. 3. Let G be a graph, (X, ϕ) be a Smale space and ρ : (ΣG , σ) → (X, ϕ) be a regular, s-resolving factor map. 1 for the map δn : (ΣN (ρ), σ) → (ΣN −1 (ρ), σ). 4. Let G and H be graphs and (X, ϕ) be a Smale space.

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. Toward that end, we make the following additional definition. 10. (1) For each L ≥ 0, let ρL, : ΣL,0 (π) → YL (πs ) be the map defined by ρL, (y0 , . . , yL , z0 ) = (y0 , .

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). For each L, M ≥ 0, we define ΣL,M (π) = {(y0 , . . , yL , z0 , . . , zM ) | yl ∈ Y, zm ∈ Z, πs (yl ) = πu (zm ), 0 ≤ l ≤ L < 0 ≤ m ≤ M } For convenience, we also let Σ(π) = Σ0,0 (π), which is simply the fibred product of the spaces Y and Z. We let ρu (y, z) = y and ρs (y, z) = z denote the usual maps from Σ(π) to Y and Z respectively.

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A homology theory for Smale spaces by Ian F. Putnam

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