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

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 ) , 2

That is, we have a local picture: ✻ X u ( x, ǫ ) � ❅ x ❅ � ✇ � ❅ X s ( x, ǫ ) ❅ � per n ( X, ϕ ) = # { x ∈ X | ϕ n ( x ) = x } 3

Theorem 1 (Manning) . For a Smale space ( X, ϕ ) , its Artin-Mazur zeta function   ∞ per n ( X, ϕ ) t n � ζ ϕ ( t ) = exp   n n =1 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 per n ( X, ϕ ) ? This talk: Yes 4

Example The linear map � � 1 1 : R 2 → R 2 A = 1 0 is hyperbolic. Let γ > 1 be the Golden mean, ( γ, 1) A = γ ( γ, 1) − γ − 1 ( − 1 , γ ) ( − 1 , γ ) A = R 2 is not compact, instead we use X = R 2 / Z 2 As det ( A ) = − 1, A induces a map with the same local structure, but is a Smale space. Stable: R , Unstable: R 5

Example : Shifts of finite type (SFTs) Let G = ( G 0 , G 1 , i, t ) be a finite directed graph. Then we have the space of bi-infinite paths and the 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 metric d ( e, f ) = 2 − k , where k ≥ 0 is the least integer where ( e − k , e k ) � = ( f − k , f k ). 6

Example with G 0 = { v, w } e such that . . . e − 2 e − 1 e 0 . t ( e 0 ) = i ( e 1 ) = v s.t. t ( e 0 ) = v .e 1 e 2 e 3 · · · s.t. i ( e 1 ) = v e such that . . . e − 2 e − 1 e 0 . t ( e 0 ) = i ( e 1 ) = w s.t. t ( e 0 ) = w .e 1 e 2 e 3 · · · s.t. i ( e 1 ) = w 7

Example Let X 0 = D × S 1 , be the solid torus and define an injection ϕ 0 : X 0 → X 0 : ϕ 0 is not a homeomorphism, instead we use X = ∩ n ≥ 1 ϕ n 0 ( X 0 ) ϕ = ϕ 0 | X . Stable: Cantor, Unstable: R . 8

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. 9

Krieger’s invariants for SFT’s Using modern terminology, Krieger looked the equivalence relations of stable and unstable equiv- alence for (Σ G , σ ): R s { ( x, y ) | lim n → + ∞ d ( σ n ( x ) , σ n ( y )) = 0 } = = right-tail equivalence R u { ( x, y ) | lim n → + ∞ 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 D s (Σ G , σ ) K 0 ( C ∗ ( R s )) , = D u (Σ G , σ ) K 0 ( C ∗ ( R u )) . = = lim Z N A G → Z N A G D s (Σ G , σ ) ∼ − − → · · · where N = # G 0 , A G = adjacency matrix of G. The automorphism σ ∗ is multiplication by A G . 10

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. 11

Manning: Keep track of when π is N -to-1, for various values of N . 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 and S N +1 acts on Σ N ( π ). Manning used the periodic point data from the sequence Σ N ( π ) (with the action of S N +1 ) to compute per n ( X, f ). This is extremely reminiscent of using the nerve of an open cover to compute homology of a compact manifold. 12

H ∗ ( M ) H ∗ ( X, ϕ ) ? ˇ ’good’ open cover Bowen’s Theorem π : (Σ , σ ) → ( X, ϕ ) U 1 , . . . , U I multiplicities multiplicities U i 0 ∩ · · · ∩ U i N � = ∅ Σ N ( π ) groups groups C N generated by D s (Σ N ( π )) alt U i 0 ∩ · · · ∩ U i N � = ∅ boundary maps boundary maps ∂ N ( U i ∩ U j ) = U j − U i ? ? 13

The problem: For 0 ≤ n ≤ N , let δ n : Σ N ( π ) → Σ N − 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 ∗ : D s (Σ , σ ) → D s (Σ ′ , σ ) between Krieger’s invariants. But this problem is well-understood in sym- bolic dynamics ... 14

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 ) → X s ( 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 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. 15

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. 16

� � � � 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 5 (Better Bowen) . If ( X, ϕ ) is a non- wandering Smale space, then there exists an s/u -bijective pair, π = ( Y, ψ, π s , Z, ζ, π u ) . Like the SFT in Bowen’s Theorem, this is not unique. The fibred product is a SFT: Σ = { ( y, z ) ∈ Y × Z | π s ( y ) = π u ( z ) } . ( Y, ψ ) ρ u π s (Σ , σ ) ( X, ϕ ) ρ s π u ( Z, ζ ) 17

Adapting Manning’s idea, for L, M ≥ 0, we de- fine 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. 18

� � � � � � � � � � � � � � � � � � 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 ) . 19

Theorem 6. The groups H s N ( π ) depend on ( X, ϕ ) , but not the choice of s/u -bijective pair π = ( Y, ψ, π s , Z, ζ, π u ) . From now on, we write H s N ( X, ϕ ). Theorem 7. The functor H s ∗ ( X, ϕ ) is covariant for s -bijective factor maps, contravariant for u - bijective factor maps. Theorem 8. The groups H s N ( X, ϕ ) are all finite rank and non-zero for only finitely many N ∈ Z . 20

Theorem 9 (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 } = = per p ( X, ϕ ) 21

Example: Shifts of finite type If ( X, ϕ ) = (Σ , σ ), then Y = Σ = Z is an s/u - bijective pair. H s D s (Σ , σ ) , 0 (Σ , σ ) = H s N (Σ , σ ) = 0 , N � = 0 . Example: 2 ∞ -solenoid [Amini, P., Saeidi Gho- likandi] ∼ H s 0 ( X, ϕ ) Z [1 / 2] , = ∼ H s 1 ( X, ϕ ) Z , = H s N ( X, ϕ ) = 0 , N � = 0 , 1 Generalized 1-solenoids (Williams, Yi, Thom- sen): Amini, P, Saeidi Gholikandi. 22

Example: 2 -torus [Bazett-P.]: � � 1 1 : R 2 / Z 2 → R 2 / Z 2 1 0 H s ϕ s N N ( X, ϕ ) − 1 1 Z � � 1 1 Z 2 0 1 0 1 − 1 . Z 23