A homology theory for Smale spaces Ian F. Putnam, University of - - PDF document

A homology theory for Smale spaces

Ian F. Putnam, University of Victoria

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Smale spaces (D. Ruelle) (X, d) compact metric space, ϕ : X → X homeomorphism, 0 < λ < 1, For x in X and ǫ > 0 and small, there is a local stable set Xs(x, ǫ) and a local unstable set Xu(x, ǫ): 1. Xs(x, ǫ) × Xu(x, ǫ) is homeomorphic to a neighbourhood of x,

• 2. ϕ-invariance,

3. d(ϕ(y), ϕ(z)) ≤ λd(y, z), y, z ∈ Xs(x, ǫ), d(ϕ−1(y), ϕ−1(z)) ≤ λd(y, z), y, z ∈ Xu(x, ǫ),

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That is, we have a local picture:

✇ ✻

x Xu(x, ǫ) Xs(x, ǫ)

• ❅

❅

• ❅
• ❅

pern(X, ϕ) = #{x ∈ X | ϕn(x) = x}

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Theorem 1 (Manning). For a Smale space (X, ϕ), its Artin-Mazur zeta function ζϕ(t) = exp

 

∞

• n=1

pern(X, ϕ) n tn

 

is rational. Question 2 (Bowen). Is there a homological interpretation of this? Is there a homology theory H∗(X, ϕ) providing a Lefschetz formula which computes pern(X, ϕ)? This talk: Yes

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Example The linear map A =

• 1

1 1

• : R2 → R2

is hyperbolic. Let γ > 1 be the Golden mean, (γ, 1)A = γ(γ, 1) (−1, γ)A = −γ−1(−1, γ) R2 is not compact, instead we use X = R2/Z2 As det(A) = −1, A induces a map with the same local structure, but is a Smale space. Stable: R, Unstable: R

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Example : Shifts of finite type (SFTs) Let G = (G0, G1, i, t) be a finite directed graph. Then we have the space of bi-infinite paths and the shift map: ΣG = {(ek)∞

k=−∞ | ek ∈ G1,

i(ek+1) = t(ek), for all n} σ(e)k = ek+1, ”left shift” The metric d(e, f) = 2−k, where k ≥ 0 is the least integer where (e−k, ek) = (f−k, fk).

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Example with G0 = {v, w} . . . e−2e−1e0. s.t. t(e0) = v e such that .e1e2e3 · · · s.t. i(e1) = v t(e0) = i(e1) = v . . . e−2e−1e0. s.t. t(e0) = w .e1e2e3 · · · s.t. i(e1) = w e such that t(e0) = i(e1) = w

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Example Let X0 = D × S1, be the solid torus and define an injection ϕ0 : X0 → X0: ϕ0 is not a homeomorphism, instead we use X = ∩n≥1ϕn

0(X0)

ϕ = ϕ0|X. Stable: Cantor, Unstable: R.

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Recall the problem: find a homology theory for Smale spaces. Step 1: Find the invariant for shifts of finite type: Wolfgang Krieger (1980). (There is also another by Bowen and Franks.) Step 2: Extend it to all Smale spaces.

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Krieger’s invariants for SFT’s Using modern terminology, Krieger looked the equivalence relations of stable and unstable equiv- alence for (ΣG, σ): Rs = {(x, y) | limn→+∞d(σn(x), σn(y)) = 0} = right-tail equivalence Ru = {(x, y) | limn→+∞d(σ−n(x), σ−n(y)) = 0} = left-tail equivalence and constructed their groupoid C∗-algebras. These are each AF-algebras with stationary Bratteli diagrams and looked at Ds(ΣG, σ) = K0(C∗(Rs)), Du(ΣG, σ) = K0(C∗(Ru)). Ds(ΣG, σ) ∼ = lim ZN AG − → ZN AG − → · · · where N = #G0, AG = adjacency matrix of G. The automorphism σ∗ is multiplication by AG.

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To extend the invariant from SFT to Smale spaces: Theorem 3 (Bowen). For a (non-wandering) Smale space, (X, ϕ), there exists a SFT (Σ, σ) and π : (Σ, σ) → (X, ϕ), with π ◦ σ = ϕ ◦ π, continuous, surjective and finite-to-one. Since π is finite-to-one it provides a surjec- tive (not injective) map between the periodic points. (Σ, σ) is not unique.

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Manning: Keep track of when π is N-to-1, for various values of N. For N ≥ 0, define ΣN(π) = {(e0, e1, . . . , eN) | π(en) = π(e0), 0 ≤ n ≤ N}. For all N ≥ 0, (ΣN(π), σ) is also a shift of finite type and SN+1 acts on ΣN(π). Manning used the periodic point data from the sequence ΣN(π) (with the action of SN+1) to compute pern(X, f). This is extremely reminiscent of using the nerve

• f an open cover to compute homology of a

compact manifold.

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ˇ H∗(M) H∗(X, ϕ) ? ’good’ open cover Bowen’s Theorem U1, . . . , UI π : (Σ, σ) → (X, ϕ) multiplicities multiplicities Ui0 ∩ · · · ∩ UiN = ∅ ΣN(π) groups groups CN generated by Ds(ΣN(π))alt Ui0 ∩ · · · ∩ UiN = ∅ boundary maps boundary maps ∂N(Ui ∩ Uj) = Uj − Ui ? ?

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The problem: For 0 ≤ n ≤ N, let δn : ΣN(π) → ΣN−1(π) be the map which deletes entry n. This is a nice map between the dynamical systems. Unfortunately, a map ρ : (Σ, σ) → (Σ′, σ) between shifts of finite type does not always induce a group homomorphism ρ∗ : Ds(Σ, σ) → Ds(Σ′, σ) between Krieger’s invariants. But this problem is well-understood in sym- bolic dynamics ...

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A map π : (Y, ψ) → (X, ϕ) map between Smale spaces is π is s-bijective if, for all y in Y π : Y s(y, ǫ) → Xs(π(y), ǫ′) is a local homeomorphism. Theorem 4. Let π : (Σ, σ) → (Σ′, σ) be a fac- tor map between SFT’s. If π is s-bijective, then there is a map πs : Ds(Σ, σ) → Ds(Σ′, σ). If π is u-bijective, then there is a map πs∗ : Ds(Σ′, σ) → Ds(Σ, σ). Bowen’s π : (Σ, σ) → (X, ϕ) is not s-bijective

• r u-bijective if X is a torus, for example.

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A better Bowen’s Theorem Let (X, ϕ) be a Smale space. We look for a Smale space (Y, ψ) and a factor map πs : (Y, ψ) → (X, ϕ) satisfying:

• 1. πs is s-bijective,
• 2. dim(Y u(y, ǫ)) = 0.

That is, Y u(y, ǫ) is totally disconnected, while Y s(y, ǫ) is homeomorphic to Xs(πs(y), ǫ). This is a “one-coordinate” version of Bowen’s Theorem.

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Similarly, we look for a Smale space (Z, ζ) and a factor map πu : (Z, ζ) → (X, ϕ) satisfying dim(Zs(z, ǫ)) = 0, and πu is u-bijective. We call π = (Y, ψ, πs, Z, ζ, πu) a s/u-bijective pair for (X, ϕ). Theorem 5 (Better Bowen). If (X, ϕ) is a non- wandering Smale space, then there exists an s/u-bijective pair, π = (Y, ψ, πs, Z, ζ, πu). Like the SFT in Bowen’s Theorem, this is not unique. The fibred product is a SFT: Σ = {(y, z) ∈ Y × Z | πs(y) = πu(z)}. (Y, ψ)

πs

• (Σ, σ)

ρu

• ρs
• (X, ϕ)

(Z, ζ)

πu

• 17

Adapting Manning’s idea, for L, M ≥ 0, we de- fine ΣL,M(π) = {(y0, . . . , yL, z0, . . . , zM) | yl ∈ Y, zm ∈ Z, πs(yl) = πu(zm)}. Each of these is a SFT. Moreover, the maps δl, : ΣL,M → ΣL−1,M, δ,m : ΣL,M → ΣL,M−1 which delete yl and zm are s-bijective and u- bijective, respectively. This is the key point! We have avoided the issue which caused our earlier attempt to get a chain complex to fail.

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We get a double complex: Ds(Σ0,2)alt

• Ds(Σ1,2)alt
• Ds(Σ2,2)alt
• Ds(Σ0,1)alt
• Ds(Σ1,1)alt
• Ds(Σ2,1)alt
• Ds(Σ0,0)alt
• Ds(Σ1,0)alt
• Ds(Σ2,0)alt
• ∂s

N :

⊕L−M=NDs(ΣL,M)alt → ⊕L−M=N−1Ds(ΣL,M)alt ∂s

N =

L

l=0(−1)lδs l, + M+1 m=0 (−1)m+Mδs∗ ,m

Hs

N(π) = ker(∂s N)/Im(∂s N+1).

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Theorem 6. The groups Hs

N(π) depend on

(X, ϕ), but not the choice of s/u-bijective pair π = (Y, ψ, πs, Z, ζ, πu). From now on, we write Hs

N(X, ϕ).

Theorem 7. The functor Hs

∗(X, ϕ) is covariant

for s-bijective factor maps, contravariant for u- bijective factor maps. Theorem 8. The groups Hs

N(X, ϕ) are all finite

rank and non-zero for only finitely many N ∈ Z.

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Theorem 9 (Lefschetz Formula). Let (X, ϕ) be any non-wandering Smale space and let p ≥ 1.

• N∈Z

(−1)N Tr[(ϕs)−p : Hs

N(X, ϕ) ⊗ Q

→ Hs

N(X, ϕ) ⊗ Q]

= #{x ∈ X | ϕp(x) = x} = perp(X, ϕ)

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Example: Shifts of finite type If (X, ϕ) = (Σ, σ), then Y = Σ = Z is an s/u- bijective pair. Hs

0(Σ, σ)

= Ds(Σ, σ), Hs

N(Σ, σ)

= 0, N = 0. Example: 2∞-solenoid [Amini, P., Saeidi Gho- likandi] Hs

0(X, ϕ)

∼ = Z[1/2], Hs

1(X, ϕ)

∼ = Z, Hs

N(X, ϕ)

= 0, N = 0, 1 Generalized 1-solenoids (Williams, Yi, Thom- sen): Amini, P, Saeidi Gholikandi.

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Example: 2-torus[Bazett-P.]:

• 1

1 1

• : R2/Z2 → R2/Z2

N Hs

N(X, ϕ)

ϕs −1 Z 1 Z2

• 1

1 1

• 1

Z −1.

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