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 really is in accordance with parts. the 1st is a far better model of Bowen's outcome 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 crew invariant for shifts of finite style. He proves a Lefschetz formulation which relates the variety of periodic issues of the method 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 by means of Bowen within the Seventies

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Extra resources for A homology theory for Smale spaces

Example text

The factor map sends (a, ±∞) and (b, ±∞) to ±∞ and is the identity on the other points. Fix n in Z and notice that E = {(a, x) | −∞ ≤ x ≤ n} is in CO s (ΣH , σ), while its image under the factor map is not in CO s (ΣG , σ). Hence, the ﬁrst part of the result above fails with the hypothesis of s-bijective replaced by s-resolving. 4. 4. Let D denote the group Z2 with lexicographic order. We have Ds (ΣH , σ) ∼ = D ⊕ D, while Ds (ΣH , σ) ∼ = D. There does indeed exist a well-deﬁned homomorphism from the former to the latter, but it is not induced dynamically.

Suppose that G is a graph. 6 that ΣsG (e, 2−K ) = {f ∈ ΣG | ek = f k , for all k ≥ 1 − K}. It is easy to see that such sets are in CO s (ΣG , σ) and that if e, e are in ΣG and satisfy i(e1−K ) = i(e 1−K ) then ΣsG (e, 2−K ) ∼ ΣsG (e , 2−K ). Moreover, every set in CO s (ΣG , σ) may be expressed as a ﬁnite union of such sets (allowing both e and K to vary). In a similar way, we let CO u (Σ, σ) be the collection of all non-empty, compact, open subsets of Σu (e). Let ∼ be the smallest equivalence relation on CO u (Σ, σ) such that E ∼ F if [E, F ] = F and [F, E] = E (meaning that both sets are deﬁned) and such that E ∼ F if and only if σ(E) ∼ σ(F ).

13, since Y1 (π) is the ﬁbred product of Y with itself. 13 and observing that YN +1 (π) is the ﬁbred product of Y with YN (π). The ﬁnal part of this section deals with the special case of maps between two shifts of ﬁnite type. We begin with the following easy result. 28 2. 15. Let π : (ΣG , σ) → (X, ϕ) be a regular map. Suppose N, k ≥ 1 and p is in GkN (π). (1) If π is s-resolving and tk (pi ) = tk (pj ) for some i, j, then pi = pj . (2) If π is u-resolving and ik (pi ) = ik (pj ) for some i, j, then pi = pj .