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 1

Hyperbolicity An invertible linear map T : R d → R d is hyper- bolic if R d = E s ⊕ E u , T -invariant, C > 0 , 0 < λ < 1, � T n v � ≤ Cλ n � v � , v ∈ E s , n ≥ 1 � T − n v � ≤ Cλ n � v � , v ∈ E u , n ≥ 1 Same definition replacing R d by a vector bundle (over compact space). M compact manifold, ϕ : M → M diffeomor- phism is Anosov if Dϕ : TM → TM is hyper- bolic. Smale: M, ϕ Axiom A : replace TM above by TM | NW ( ϕ ) = E s ⊕ E u , where NW ( ϕ ) is the set of non-wandering points. But NW ( ϕ ) is usually a fractal, not a submanifold. 2

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 X s ( x, ǫ ) and a local unstable set X u ( x, ǫ ): X s ( x, ǫ ) × X u ( x, ǫ ) is homeomorphic to a 1. neighbourhood of x , 2. ϕ -invariance, 3. y, z ∈ X s ( x, ǫ ) , d ( ϕ ( y ) , ϕ ( z )) ≤ λd ( y, z ) , d ( ϕ − 1 ( y ) , ϕ − 1 ( z )) y, z ∈ X u ( x, ǫ ) , ≤ λd ( y, z ) , 3

That is, we have a local picture: ✻ X u ( x, ǫ ) � ❅ x ❅ � ✇ X s ( x, ǫ ) � ❅ ❅ � Global stable and unstable sets: X s ( x ) n → + ∞ d ( ϕ n ( x ) , ϕ n ( y )) = 0 } = { y | lim X u ( x ) n → + ∞ d ( ϕ − n ( x ) , ϕ − n ( y )) = 0 } = { y | lim These are equivalence relations. X s ( x, ǫ ) ⊂ X s ( x ), X u ( x, ǫ ) ⊂ X u ( x ). 4

Example 1 � � 1 1 The linear map A = is hyperbolic. Let 1 0 γ > 1 be the Golden mean, ( γ, 1) A = γ ( γ, 1) − γ − 1 ( − 1 , γ ) ( − 1 , γ ) A = As det ( A ) = − 1, it induces a homeomorphism of R 2 / Z 2 which is Anosov. X s and X u are Kronecker foliations with lines of slope − γ − 1 and γ . 5

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Example 3: Shifts of finite type (SFTs) Let G = ( G 0 , G 1 , i, t ) be a finite directed graph. Then we have the shift space and shift map: k = −∞ | e k ∈ G 1 , { ( e k ) ∞ Σ G = i ( e k +1 ) = t ( e k ) , for all n } σ ( e ) k e k +1 , ”left shift” = The local product structure is given by Σ s ( e, 1) = { ( . . . , ∗ , ∗ , ∗ , e 0 , e 1 , e 2 , . . . ) } Σ u ( e, 1) = { ( . . . , e − 2 , e − 1 , e 0 , ∗ , ∗ , ∗ , . . . ) } 7

Smales spaces have a large supply of periodic points and it is interesting to count them. Adjacency matrix of G : G 0 = { 1 , 2 , . . . , N } , A G is N × N with ( A G ) i,j = #edges from i to j Theorem 1. Let A G be the adjancency matrix of the graph G . For any p ≥ 1 , we have # { e ∈ Σ G | σ p ( e ) = e } = Tr ( A p G ) . This is reminiscent of the Lefschetz fixed-point formula for smooth maps of compact mani- folds. Question 2. Is the right hand side actually the result of σ acting on some homology theory of (Σ G , σ ) ? Positive answers by Bowen-Franks and Krieger. 8

Krieger’s invariants for SFT’s W. Krieger defined invariants, which we de- note by D s (Σ G , σ ) , D u (Σ G , σ ), for shifts of fi- nite type by considering stable and unstable equivalence as groupoids and taking its groupoid C ∗ -algebra: K 0 ( C ∗ ( X s )) , K 0 ( C ∗ ( X s )) In this case, these are both AF-algebras and D s (Σ G , σ ) = lim Z N A G → Z N A G − − → · · · (For the unstable, replace A G with A T G .) Each comes with a canonical automorphism. Returning to Smale spaces . . . 9

Bowen’s Theorem 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. First, this means that SFT’s have a special place among Smale spaces. Secondly, one can try to understand ( X, ϕ ) by investigating (Σ , σ ). For example, they will have the same entropy. Of course, (Σ , σ ) is not unique. A. Manning used Bowen’s Theorem to pro- vide a formula counting the number of periodic points for ( X, ϕ ). 10

For N ≥ 0, define Σ N ( π ) = { ( e 0 , e 1 , . . . , e N ) | π ( e n ) = π ( e 0 ) , 0 ≤ n ≤ N } . For all N ≥ 0, (Σ N ( π ) , σ ) is also a shift of finite type. Observe that S N +1 acts on Σ N ( π ). Theorem 4 (Manning) . For a non-wandering Smale space ( X, ϕ ) , (Σ , σ ) as above and p ≥ 1 , we have # { x ∈ X | ϕ p ( x ) = x } N ( − 1) N Tr ( σ p ∗ : D s (Σ N ( π )) alt = � → D s (Σ N ( π )) alt ) . Question 5 (Bowen) . Is there a homology the- ory for Smale spaces H ∗ ( X, ϕ ) which provides a Lefschetz formula, counting the periodic points? In fact, the groups D s (Σ N ( π )) alt appear to be giving a chain complex. 11

Idea: for 0 ≤ n ≤ N , let δ n : Σ N ( π ) → Σ N − 1 ( π ) be the map which deletes entry n . Let ( δ n ) ∗ : D s (Σ N ( π )) alt → D s (Σ N − 1 ( π )) alt be the induced map and ∂ = � N n =0 ( − 1) n ( δ n ) ∗ to make a chain complex. This is wrong: a map ρ : (Σ , σ ) → (Σ ′ , σ ) between shifts of finite type does not always in- duce a group homomorphism between Krieger’s invariants. While it is true that ρ will map R s (Σ) to R s (Σ ′ ) the functorial properties of the construction of groupoid C ∗ -algebras is subtle. 12

Let π : ( Y, ψ ) → ( X, ϕ ) be a factor map be- tween Smale spaces. For every y in Y , we have π ( Y s ( y )) ⊆ X s ( π ( y )). Definition 6. π is s -bijective if π : Y s ( y ) → X s ( π ( y )) is bijective, for all y . Theorem 7. If π is s -bijective then π : Y s ( y, ǫ ) → X s ( π ( y ) , ǫ ′ ) is a local homeomorphism. Theorem 8. Let π : (Σ , σ ) → (Σ ′ , σ ) be a fac- tor map between SFT’s. If π is s -bijective, then there is a map π s : D s (Σ , σ ) → D s (Σ ′ , σ ) . If π is u -bijective, then there is a map π s ∗ : D s (Σ ′ , σ ) → D s (Σ , σ ) . Bowen’s π : (Σ , σ ) → ( X, ϕ ) is not s -bijective or u -bijective if X is a torus, for example. 13

A better Bowen’s Theorem Let ( X, ϕ ) be a Smale space. We look for a Smale space ( Y, ψ ) and a factor map π s : ( Y, ψ ) → ( X, ϕ ) satisfying: 1. π s 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 X s ( π s ( y ) , ǫ ). This is a “one-coordinate” version of Bowen’s Theorem. 14

� � Similarly, we look for a Smale space ( Z, ζ ) and a factor map π u : ( Z, ζ ) → ( X, ϕ ) satisfying dim ( Z s ( z, ǫ )) = 0, and π u is u -bijective. We call π = ( Y, ψ, π s , Z, ζ, π u ) a s/u -bijective pair for ( X, ϕ ). Theorem 9. If ( X, ϕ ) is a non-wandering Smale space, then there exists an s/u -bijective pair. Consider the fibred product: Σ = { ( y, z ) ∈ Y × Z | π s ( y ) = π u ( z ) } with Σ ρ u ρ s � � � � � � � � � � � � � � � � � Y Z � � � � � � � � � π s π u � � � � � � � � X ρ s ( y, z ) = z is s -bijective, ρ u ( y, z ) = y is u - bijective. Hence, Σ is a SFT. 15

For L, M ≥ 0, we define Σ L,M ( π ) = { ( y 0 , . . . , y L , z 0 , . . . , z M ) | y l ∈ Y, z m ∈ Z, π s ( y l ) = π u ( z m ) } . Each of these is a SFT. Moreover, the maps δ l, : Σ L,M → Σ L − 1 ,M , δ ,m : Σ L,M → Σ L,M − 1 which delete y l and z m 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. 16

� � � � � � � � � � � � � � � � � � We get a double complex: D s (Σ 0 , 2 ) alt D s (Σ 1 , 2 ) alt D s (Σ 2 , 2 ) alt D s (Σ 0 , 1 ) alt D s (Σ 1 , 1 ) alt D s (Σ 2 , 1 ) alt D s (Σ 0 , 0 ) alt D s (Σ 1 , 0 ) alt D s (Σ 2 , 0 ) alt ∂ s ⊕ L − M = N D s (Σ L,M ) alt N : ⊕ L − M = N − 1 D s (Σ L,M ) alt → l, + � M +1 ∂ s � L l =0 ( − 1) l δ s m =0 ( − 1) m + M δ s ∗ N = ,m H s N ( π ) = ker( ∂ s N ) /Im ( ∂ s N +1 ) . 17

Recall: beginning with ( X, ϕ ), we select an s/u -bijective pair π = ( Y, ψ, π s , Z, ζπ u ) construct the double complex and compute H s N ( π ). Theorem 10. The groups H s N ( π ) do not de- pend on the choice of s/u -bijective pair π . From now on, we write H s N ( X, ϕ ). Theorem 11. The functor H s ∗ ( X, ϕ ) is covari- ant for s -bijective factor maps, contravariant for u -bijective factor maps. Theorem 12. The groups H s N ( X, ϕ ) are all fi- nite rank and non-zero for only finitely many N ∈ Z . 18

We can regard ϕ : ( X, ϕ ) → ( X, ϕ ), which is both s and u -bijective and so induces an auto- morphism of the invariants. Theorem 13. (Lefschetz Formula) Let ( X, ϕ ) be any non-wandering Smale space and let p ≥ 1 . Tr [( ϕ s ) p : ( − 1) N H s � N ( X, ϕ ) ⊗ Q N ∈ Z H s N ( X, ϕ ) ⊗ Q ] → # { x ∈ X | ϕ p ( x ) = x } = 19

� � � � � � � � � � � � � � � � � � Example 1: Shifts of finite type If ( X, ϕ ) = (Σ , σ ), then Y = Σ = Z is an s/u - bijective pair. The double complex D s a is: 0 0 0 0 0 0 D s (Σ) 0 0 and H s 0 (Σ , σ ) = D s (Σ) and H s N (Σ , σ ) = 0 , N � = 0. 20