A signed version of Putnam's homology theory: Lefschetz and zeta functions
aa r X i v : . [ m a t h . D S ] D ec A signed version of Putnam’s homology theory:Lefschetz and zeta functions
Robin J Deeley
Abstract
A signed version of Putnam homology for Smale spaces is introduced.Its definition, basic properties and associated Lefschetz theorem are outlined. Inparticular, zeta functions associated to an Axiom A diffeomorphism are compared.
Let ( M , f ) be an Axiom A diffeomorphism [10]. Then there are two natural zetafunctions associated to ( M , f ) , the dynamical zeta function and the homologicalzeta function, see [10, Section I.4]. The former is defined as follows: z dym ( s ) : = exp (cid:229) n ≥ N n n t n ! where N n is the cardinality of the set of points with period n . The definition of latteris z hom ( s ) : = exp (cid:229) n ≥ ˜ N n n t n ! where ˜ N n is obtained by counting the points of period n with “sign” (see Example 8or [10, Section I.4] for further details).Both these functions extend meromorphically to rational functions. For the for-mer, this is an important theorem of Manning [7]. For the latter, it is a corollary ofthe Lefschetz fixed point theorem. Based on Manning’s result, Bowen asked whetherthere exists a homology theory for basic sets of an Axiom A diffeomorphism alongwith an associated Lefschetz theorem that implies that the dynamical zeta func- Robin DeeleyDepartment of Mathematics, University of Hawaii, 2565 McCarthy Mall, Keller 401A, HonoluluHI 96822 e-mail: [email protected] tion is a rational function in the same way the classical Lefschetz theorem impliesthat the homological zeta function is a rational function. Recently, Ian Putnam con-structed such a homology theory and proved the relevant Lefschetz theorem [8] (inparticular see [8, Section 6]). For more on the relationship between zeta functions,homology, and Lefschetz theorems see [10, Section I.4] or [8, Section 6.1] for briefintroductions or [5] and references therein for more details.Putnam’s homology theory is defined using the framework of Smale spaces.Smale spaces were introduced by Ruelle [9]. The reader who is unfamiliar withthem can assume that any Smale space in the present paper is either the nonwan-dering set or a basic set of an Axiom A diffeomorphism. The precise definition of aSmale space is given in Section 2.It is important to note that the dynamical zeta function of an Axiom A diffeo-morphism depends only on its restriction to the nonwandering set, but this is not thecase for the homological zeta function. In particular, the two zeta functions definedabove are not in general equal and as such the classical homology of M and Putnam’shomology of the nonwandering set of M are (again in general) not isomorphic.The modest goal of the present paper is to outline the construction of a homologytheory defined in the same spirit as Putnam’s homology, but whose associated Lef-schetz theorem is more closely related to the classical Lefschetz theorem; it countsperiodic points with “sign”, see Theorem 5 for the precise statement. This goal isachieved by considering signed Smale spaces. By definition, a signed Smale spaceis a Smale space along with a continuous map to {− , } , which is called a signfunction. Then, by following Putnam’s constructions in [8] quite closely but withthis additional sign function, one obtains a new “signed version” of Putnam’s ho-mology. In the case of an Axiom A diffeomorphism, the signed homology theory ofthe nonwandering set with a particular sign function is more closely related (in par-ticular through the associated Lefschetz theorem) to the standard homology of themanifold, at least in particular situations, see Theorem 6 and Example 10. The no-tion of signed Smale space is based on work of Bowen, see in particular [1, Theorem2]. If the Smale space is connected the only possible sign functions are constantand the signed homology is essentially the same as Putnam’s homology. However,typically the nonwandering set of an Axiom A diffeomorphism is not connected andthis can also occur for basic sets (e.g., shifts of finite type).I have assumed the reader is familiar with Putnam’s monograph [8] and Bowen’spaper [1]. In particular, see [1] for more on filtrations and the no-cycle condition.Also, the reader should be warned that there are many definitions and a numberproofs are omitted. Most notable among these are Theorems 4 and 5. Although,proofs of these theorems are long, the reader familiar with the proofs in [8] willlikely see how they are proved. In particular, for Theorem 5, one follows almostverbatim the construction (which is based on Manning’s proof in [7]) in [8, Section6]. Detailed proofs of these theorems will appear elsewhere. signed version of Putnam’s homology theory: Lefschetz and zeta functions 3 Definition 1.
A Smale space ( X , j ) consists of a compact metric space ( X , d ) and ahomeomorphism j : X → X such that there exist constants e X > , < l < { ( x , y ) ∈ X × X | d ( x , y ) ≤ e X } 7→ [ x , y ] ∈ X satisfying the following axioms:B1 [ x , x ] = x ,B2 [ x , [ y , z ]] = [ x , z ] ,B3 [[ x , y ] , z ] = [ x , z ] , andB4 j [ x , y ] = [ j ( x ) , j ( y )] ;where in these axioms, x , y , and z are in X and in each axiom both sides are assumedto be well-defined. In addition, ( X , j ) is required to satisfyC1For x , y ∈ X such that [ x , y ] = y , we have d ( j ( x ) , j ( y )) ≤ l d ( x , y ) andC2For x , y ∈ X such that [ x , y ] = x , we have d ( j − ( x ) , j − ( y )) ≤ l d ( x , y ) .The map [ · , · ] in the definition of a Smale space is called the bracket map; it isunique (provided it exists). Example 1. If ( M , f ) is an Axiom A diffeomorphism, then the restriction of f to thenonwandering set is a Smale space and likewise the restriction of f to a basic set isalso a Smale space. The bracket map in the definition of a Smale space is, in thesecases, given by the canonical coordinates.An important class of Smale spaces are the shifts of finite type. They can be definedas follows. Let G = ( G , G , i , t ) be a directed graph; that is, G and G are finitesets called the set of vertices and the set of edges and each edge e ∈ G is givenby a directed edge from i ( e ) ∈ G to t ( e ) ∈ G , see [8, Definition 2.2.1] for furtherdetails.From G a dynamical system is constructed by taking S G : = { ( g j ) j ∈ Z | g j ∈ G and t ( g j ) = i ( g j + ) for each j ∈ Z } with the homeomorphism, s : S G → S G given by left sided shift. Then, see forexample [8], ( S G , s ) is a Smale space and one can define a shift of finite type to beany dynamical system that is conjugate to ( S G , s ) for some graph G . Often we willdrop the G from the notation and denote a shift of finite type by ( S , s ) .From G and k ≥
2, one can obtain a higher block presentation by constructinganother graph G k whose edges are paths in G of length k and whose vertices arepaths in G of length k −
1; for the precise details see [8, Definition 2.2.2].
Robin J Deeley
Definition 2. A signed Smale space is a Smale space ( X , j ) along with a continuousmap D X : X → {− , } . Furthermore, for n ≥
1, we define D ( n ) X ( x ) = n − (cid:213) i = D X ( j i ( x )) . A signed Smale space is denoted by ( X , j , D X ) and D X is called the sign function; itis often denoted simply by D . Example 2.
Let ( W , f | W ) be a basic set of an Axiom A diffeomorphism, ( M , f ) . Weassume that the bundle E u | W can be oriented and then define D : W → {− , } asfollows: D ( x ) = (cid:26) D x ( f ) : ( E u | W ) | x → ( E u | W ) | f ( x ) preserves the orientation − D x ( f ) : ( E u | W ) | x → ( E u | W ) | f ( x ) reverses the orientation.The fact that W is hyperbolic implies that D is continuous; hence ( W , f | W , D ) is asigned Smale space.A special case of Example 2 occurs in both the statement and proof of [1, Theorem2]. Another class of examples are hyperbolic toral automorphisms: Example 3.
Let M = R / Z and f = A where A ∈ M ( Z ) with det ( A ) = ± l and l such that 0 < | l | < < | l | . This diffeomorphism is globallyhyperbolic and the nonwandering set is the entire manifold; that is, W = M .The bundle E u is isomorphic to the trivial rank one bundle. Its fiber, for exampleat the origin, is the eigenspace associated to l . One can then show that for any x ∈ M , D ( x ) = sign ( l ) . Definition 3.
Let ( S , s ) be a shift of finite type and D S : S → {− , } be a contin-uous function. Then ( S , s , D S ) is called a signed shift of finite type. Proposition 1.
Let ( S , s , D S ) be a signed shift of finite type. Then, there exists agraph G such that1. there is conjugacy h : ( S G , s ) → ( S , s ) ;2. for any ( g j ) j ∈ Z ∈ S G , ( D S ◦ h )(( g j ) j ∈ Z ) depends only on g .Proof. The first item is a possible definition of a shift of finite type. Using the factthat D S is continuous, one can obtain the second item by taking a higher blockpresentation. signed version of Putnam’s homology theory: Lefschetz and zeta functions 5 Definition 4.
Let ( S , s , D S ) be a signed shift of finite type and G is a graph whichsatisfies the conclusions of the previous theorem. Then, G (and the conjugacy h : ( S G , s ) → ( S , s ) ) is called a signed presentation of ( S , s , D S ) . We denote D S ◦ h by D S G .By assumption, for ( g j ) j ∈ Z ∈ S G , D G (( g j ) j ∈ Z ) depends only on g . As such, thefunction D G : G → {− , } defined via D G ( g ) : = D S G (( g j ) j ∈ Z ) (where g = g ) iswell-defined. Moreover, for a path g · · · g m in G and n ≤ m , we define D G , n ( g · · · g m ) = m (cid:213) j = m − n D G ( g j ) . Finally, given D G as above, we define D G k : G k → {− , } via D G k ( g g · · · g k − ) = D G ( g k − ) where g g · · · g k − is an element in G k (i.e., a path of length k in G ). This choice isbased on [8, Theorem 3.2.3 Part 1]. We use ( S G , s , D G ) to denote a signed shift offinite type with a fixed signed presentation. It is important to note that D S G and D G are related, but not the same; their domains are different. Definition 5.
Suppose ( S , s , D G ) is a signed shift of finite type with a fixed signedpresentation. Define g sG , D G : Z G → Z G as follows: for each v ∈ G , we let v (cid:229) e ∈ G , t ( e )= v i ( e ) · D G ( e ) . Furthermore, define D s D G ( G ) to be the inductive limit group: lim → ( Z G , g sG , D G ) . Example 4.
Let G be the graph with one vertex and two edges labelled by 0 and 1.Then the associated shift of finite type is the full two shift, ( S G , s ) . Furthermore, let D G : G → {− , } be the continuous map D G ( g ) = (cid:26) g = − g = . Then, in this case, D s D G ( S G ) ∼ = { } . Theorem 1. (reformulation of [1, Theorem 2])Suppose ( M , f ) is an Axiom A diffeomorphism satisfying the no-cycle condition, dim ( W s ) = , and q : = rank ( E u | W s ) . Then there exists signed shift of finite type ( S G , s , D G ) such that1. ( S G , s ) is conjugate to ( W s , f | W s ) ;2. the map g sG , D G : Z G → Z G has the same nonzero eigenvalues as the map onhomology: f | M s : H q ( M s , M s − ) → H q ( M s , M s − ) ;3. the map g sG , D G : D s D G ( G ) → D s D G ( G ) has the same nonzero eigenvalues as the mapon homology: f | M s : H q ( M s , M s − ) → H q ( M s , M s − ) ; Robin J Deeley where ( M s ) ms = is a fixed filtration associated to the basic sets, ( W s ) ms = , of ( M , f ) ;we assume it satisfies the assumptions in [1].Proof. Theorem 2 of [1] implies the existence of the signed shift of finite typesatisfying items (1) and (2) in the statement. Basic properties of inductive limitsof abelian groups imply that for any signed shift of finite type g sG , D G : D s D G ( G ) → D s D G ( G ) and g sG , D G : Z G → Z G have the same nonzero eigenvalues; item (3) fol-lows from this observation. Definition 6. (see [8, Definition 2.5.5])Suppose ( X , j ) and ( Y , y ) are Smale spaces and p : ( X , j ) → ( Y , y ) is a factor map.Then p is s-bijective (resp. u-bijective) if, for each x ∈ X , p | X s ( x ) (resp. p | X u ( x ) ) is abijection to X s ( p ( x )) (resp. X u ( p ( x )) ). Definition 7. (compare with [8, Definition 2.6.2])Suppose ( X , j , D X ) is a signed Smale space. Then a signed s/u-bijective pair is thefollowing data:1. signed Smale spaces ( Y , y , D Y ) and ( Z , z , D Z ) such that Y s ( y ) and Z u ( z ) are to-tally disconnected for each y ∈ Y and z ∈ Z ;2. s-bijective map p s : ( Y , y ) → ( X , j ) ;3. u-bijective map p u : ( Z , z ) → ( X , j ) ;such that D Y = D X ◦ p s and D Z = D X ◦ p u . Proposition 2. (compare with [8, Theorem 2.6.3])If ( X , j , D X ) is a nonwandering signed Smale space, then it has a signed s/u-bijective.Proof. By [8, Theorem 2.6.3], ( X , j ) has an s/u-bijective pair: ( Y , y , p s , Z , z , p u ) .Taking D Y : = D X ◦ p s and D Z : = D X ◦ p u leads to a signed s/u-bijective pair.For L ≥ M ≥
0, consider the Smale space (obtained via an iterated fiber productconstruction): S L , M ( p ) : = { ( y , . . . , y L , z , . . . , z M ) | p s ( y i ) = p u ( z j ) for each i , j } with s defined to be y × · · ·× y × z × · · ·× z . As the notation suggests ( S L , M ( p ) , s ) is a shift of finte type.Moreover, again for each L ≥ M ≥ D S L , M ( p ) : S L , M ( p ) → {− , } definedvia D S L , M ( p ) ( y , . . . , y L , z , . . . , z M ) = D Y ( y ) signed version of Putnam’s homology theory: Lefschetz and zeta functions 7 is a continuous map. We note that D S L , M ( p ) ( y , . . . , y L , z , . . . , z M ) is to equal D Y ( y i ) for any 0 ≤ i ≤ L and is also equal to D Z ( z j ) for any 0 ≤ j ≤ M . In particular, D S L , M ( p ) is constant on orbits of the natural action of S L + × S M + . For more detailson the action (which is the natural one) see [8, Section 5.1]. Definition 8.
Suppose that p = ( Y , y , p s , Z , z , p u ) a signed s/u-bijective pair for asigned Smale space, ( X , j , D ) . Then a graph G is a signed presentation of p if G isa presentation of p , in the sense of Definition 2.6.8 of [8], and G is also a signedpresentation, in the sense of Definition 4, of ( S , , s , D , ) . Proposition 3. (compare with [8, Theorem 2.6.9])If ( X , j , D ) is a signed Smale space and p = ( Y , y , p s , Z , z , p u ) is a signed s/u-bijective pair for ( X , j ) , then there exists a presentation of p . Moreover, if G is asigned presentation of p , then, for each L ≥ and M ≥ , G L , M is a signed presen-tation of ( S L , M ( p ) , s ) .Proof. Work of Putnam (see [8, Theorem 2.6.9]) implies that p has a presentationin the sense of [8, Definition 2.6.8]. That is, there is a graph G and conjugacy e : S , ( p ) → S G satisfying the conditions in [8, Definition 2.6.8]. Moreover, since D S , ( p ) is continuous, by possibly taking a higher block presentation of G we canensure that this presentation leads to a signed presentation of ( S , ( p ) , s , D , ) . Thesecond statement in the proposition follows as in the proof of [8, Theorem 2.6.9]and is omitted. Definition 9. (compare with Definition 5.2.1 of [8])Suppose ( X , j , D ) is a signed Smale space, p = ( Y , y , p s , Z , z , p u ) a signed s/u-bijective pair for ( X , j ) , and G is a presentation of p . Fix k ≥ L ≥
0, and M ≥ B ( G kL , M , S L × ) be the subgroup of Z G kL , M which is generated by elements ofthe following forms:a. p ∈ G kL , M with the property that p · (( a , ) = p for some non-trivial transposi-tion, a ∈ S L + ;b. p ′ = q · ( a , ) − sign ( a ) q for some q ∈ G kL , M and a ∈ S L + ;2. Q ( G kL , M , S L × ) be the quotient of Z G kL , M by B ( G kL , M ) ; we denote the quotientmap by Q ;3. A ( G kL , M , × S M + ) be { a ∈ Z G kL , M | a · ( , b ) = sign ( b ) · a for all b ∈ S M + } ; itis a subgroup of Z G kL , M . Proposition 4. (see the remark between Definitions 5.2.1 and 5.2.2 in [8])Suppose p = ( Y , y , p s , Z , z , p u ) a signed s/u-bijective pair for a signed Smale space, ( X , j , D ) and G is a signed presentation of p . Then, for each k ≥ , L ≥ , andM ≥ , g sG kL , M , D GkL , M ( B ( G kL , M , S L × )) ⊆ B ( G kL , M , S L × ) g sG kL , M , D GkL , M ( A ( G kL , M , S L × )) ⊆ A ( G kL , M , S L × ) Robin J Deeley where g sG kL , M , D GkL , M is defined in Definition 5. Definition 10. (compare with [8, Definition 5.2.2])Suppose p = ( Y , y , p s , Z , z , p u ) a signed s/u-bijective pair for a signed Smale space, ( X , j , D ) and G is a signed presentation of p . Using the previous proposition, wedefine D sQ , A , G k , D Gk ( G kL , M ) = lim → Q ( A ( G kL , M , × S M + )) , g sG kL , M , D GkL , M ! For each 0 ≤ i ≤ L , there is a map defined at the level of graphs, d si , : G kL , M → G kL − , M obtained by removing the i th entry. Likewise, for 0 ≤ j ≤ M , one has a map d s , j : G kL , M → G kL , M − that is defined by removing the L + j -entry. As in [8], these inducemaps at the level of the abelian groups introduced in the previous definition: Proposition 5. (compare with [8, Lemma 5.2.4])Suppose p = ( Y , y , p s , Z , z , p u ) a signed s/u-bijective pair for a signed Smale space, ( X , j , D ) and G is a signed presentation of p . Then, there exists k ∈ N such that d i , and d , j induced group homomorphisms: d si , : D sQ , A , G k , D Gk ( G kL , M ) → D sQ , A , G k , D Gk ( G kL − , M ) and d s ∗ , j : D sQ , A , G k , D Gk ( G kL , M ) → D sQ , A , G k , D Gk ( G kL , M + ) respectively. Definition 11. (compare with [8, Definition 5.1.7] and [8, Sections 5.2 and 5.3])Suppose p = ( Y , y , p s , Z , z , p u ) is a s/u-bijective pair for a signed Smale space, ( X , j , D ) , G is a signed presentation of p , and k is as in the statement of previousproposition. Then, we let d sQ , A , G k , D Gk ( p ) L , M : D sQ , A , D Gk ( G kL , M ) → D sQ , A , D Gk ( G kL − , M ) ⊕ D sQ , A , D Gk ( G kL , M + ) be the map L (cid:229) i = ( − ) i d si , + ( − ) L M (cid:229) j = ( − ) j d s ∗ , j . Finally, for each N ∈ Z , we let d sQ , A , G k , D Gk ( p ) N = L L − M = N d sQ , A , G k , D Gk ( p ) L , M . Theorem 2. (see [8, Sections 5.1 and 5.2])Assuming the setup of the previous definition, M L − M = N D sQ , A , D GkL , M ( G L , M ) , M L − M = N d sQ , A , G k , D Gk ( p ) L , M ! N ∈ Z signed version of Putnam’s homology theory: Lefschetz and zeta functions 9 is a complex. Definition 12. (compare with [8, Definition 5.1.11])Suppose p = ( Y , y , p s , Z , z , p u ) is a s/u-bijective pair for a signed Smale space, ( X , j , D ) and G is a signed presentation of p . We define H s ∗ ( X , j , D , p , G k ) to bethe homology of the complex M L − M = N D sQ , A , D GkL , M ( G L , M ) , M L − M = N d sQ , A , G k , D Gk ( p ) L , M ! N ∈ Z from the previous theorem. We call this the signed homology and denote it by H s ∗ ( X , j , D , p , G k ) ; it is a Z -graded abelian group. Theorem 3. (compare with [8, Theorem 5.1.12])The signed homology groups have finite rank and vanish for all but finitely manyN ∈ Z .Proof. Basic properties of inductive limits imply that the signed dimension groupshave finite rank. Hence the homology is finite rank (see for example page 131 of[8] for further details). That the homology vanishes for all but finitely many N alsofollows as in [8] pages 131-132. Theorem 4. (compare with [8, Theorem 5.5.1])The signed homology is independent of the choice of signed presentation, and thechoice of s/u-bijective pair.
Definition 13.
Suppose ( X , j , D ) is a signed Smale space. Based on the previoustheorem, for any choice of signed s/u-bijective pair, p , signed presentation G , and k large enough, we can define H sN ( X , j , D ) : = H sN ( X , j , D , p , G k ) . Proposition 6. (compare with a special case of [8, Theorem 5.4.1])Suppose ( X , j , D ) is a signed Smale space. The homeomorphism j and its inverseinduces graded group homomorphism at the level of the signed homology groups.We denote the induced maps by j s and ( j − ) s respectively.Remark 1. General functorial properties Putnam’s homology theory are nontrivial,see [3, 4, 8]. The functorial properties of the signed version are further complicatedby the requirement that the map at the level of Smale space must respect the signedstructure. The full details of these properties are not discussed here as they are notneeded for the signed Lefschetz theorem.
Example 5.
Suppose ( X , j ) is a Smale space and we take D X to be the constant func-tion one. Then, it follows from the defintions involved that H s ( X , j , D ) is Putnam’sstable homology theory. Example 6.
Suppose ( S G , s , D G ) is a signed shift of finite type. The signed homol-ogy, H sN ( S G , s , D G ) is the signed dimension group when N = N = Definition 14.
Suppose ( X , j ) is a Smale space. Then, for each n ∈ N ,Per ( X , j , n ) : = { x ∈ X | j n ( x ) = x } . Definition 15.
Suppose ( X , j , D ) is a signed Smale space. Then, the signed dynam-ical zeta function is z ( X , D ) ( z ) = exp ¥ (cid:229) n = N n ( X , j , D ) n z n ! where N n ( X , j , D ) = (cid:229) x ∈ Per ( X , j , n ) D ( n ) ( x ) . Example 7. If ( X , j , D ) is a signed Smale space with D ≡
1, then the signed dynam-cial zeta function is the dynamical zeta function (see the Introduction): z dyn ( z ) = exp ¥ (cid:229) n = | Per ( X , j , n ) | n z n ! . For more details on this case, see [10, Section I.4] (and also [8, Chapter 6] andreferences therein).
Example 8.
Suppose ( M , f ) is an Axiom A diffeomorphism, ( W , f | W ) be the restric-tion of f to the nonwandering set, and for each m ∈ Per ( W , f | W , ) , L ( m ) : = sign ( det ( I − D f ( m ) : T m ( M ) → T m ( M ))) . The Lefschetz fixed point formula implies that (cid:229) m ∈ Per ( W , f | W , ) L ( m ) = dim ( M ) (cid:229) i = ( − ) i Tr ( f ∗ : H i ( M ; R ) → H i ( M ; R )) . Moreover, by for example [5, Proposition 5.7] or [10, Section I.4], L ( m ) = ( − ) q D ( m ) where q is the rank of E u at the point m and D is as in Example 2. From this oneobtains the homological zeta function discussed in the Introduction. Theorem 5. (compare with [8, Theorem 6.1.1])For each k ∈ N , (cid:229) N ∈ Z ( − ) N Tr (cid:0) (( j − ) sN ⊗ id Q ) n (cid:1) = (cid:229) x ∈ Per ( X , j , n ) D ( n ) ( x ) where ( j − ) sN ⊗ id Q : H sN ( X , j , D ) ⊗ Q → H sN ( X , j , D ) ⊗ Q is the map on rational-ized homology induced from j − . Definition 16.
Suppose ( M , f ) is an Axiom A diffeomorphism satisfying the no-cycle condition, ( W s ) ms = are the basic sets of ( M , f ) , and ( M s ) ms = is a filtration signed version of Putnam’s homology theory: Lefschetz and zeta functions 11 associated to the basic sets that satisfies the assumptions in [1]. Then we let f even and f odd denotes the map induced by f on M n even H n ( M s , M s − ) and M n odd H n ( M s , M s − ) respectively.Likewise if ( X , j ) is a Smale space, we let j − even and j − odd denote the map inducedby j − on M n even H sn ( X , j , D ) ⊗ Q and M n odd H sn ( X , j , D ) ⊗ Q respectively. Theorem 6.
Suppose ( M , f ) is an Axiom A diffeomorphism satisfying the no-cyclecondition, and E u | W s is orientable. Then (using the notation introduced in the para-graph preceding this theorem) there exists a signed Smale space, ( X , j , D ) , suchthat(1) ( X , j ) is conjugate to ( W s , f | W s ) ;(2) (even case) if q is even, the maps j − even ⊕ f odd and j − odd ⊕ f even have the samenonzero eigenvalues or(2) (odd case) if q is odd, the maps j − even ⊕ f even and j − odd ⊕ f odd have the samenonzero eigenvalues. Corollary 1.
Suppose ( M , f ) is an Axiom A diffeomorphism, ( W , f | W ) is the non-wandering set of ( M , f ) , E u | W is orientable, and D : W → {− , } is defined as inExample 2. Let q : W → { , } be the function defined by q ( x ) = rank ( E ux ) mod .Then1. if q ≡ , then z hom ( z ) = z ( W , D ) ( z ) ;2. if q ≡ , then z hom ( z ) = / z ( W , D ) ( z ) .Proof. By definition, the signed zeta function is given by z j ( z ) = exp ¥ (cid:229) n = N n ( X , j , D ) n z n ! where N n ( X , j , D ) = (cid:229) x ∈ Per ( X , j , n ) D ( n ) ( x ) . If q ≡
0, then (cid:229) x ∈ Per ( W , f | W , n ) L ( x ) = (cid:229) x ∈ Per ( W , f | W , n ) D ( n ) ( x ) while if q ≡
1, then (cid:229) x ∈ Per ( W , f | W , n ) L ( x ) = ( − ) (cid:229) x ∈ Per ( W , f | W , n ) D ( n ) ( x ) . The result then follows.
To conclude the paper, two examples are discussed. These examples point to thepossibility of a stronger relationship between the signed version of Putnam’s ho-mology and the standard homology of the manifold associated with the Axiom Adiffeomorphism. However, such a relationship is (at this point) highly speculative.
Example 9.
Shifts of finite type
In [2], Bowen and Franks prove the following results:
Theorem 7. (reformulation of [2, Theorem 3.2]) Suppose ( M , f ) is an Axiom A dif-feomorphism satisfying the no-cycle condition, dim ( W s ) = , and q : = rank ( E u | W s ) .Then there exists signed shift of finite type ( S G , s , D G ) such that1. ( S G , s ) is conjugate to ( W s , f | W s ) ;2. the maps g sG , D G : Z G → Z G and f | M s : H q ( M s , M s − ) → H q ( M s , M s − ) are shiftequivalent;3. the maps g sG , D G : D s D G ( G ) → D s D G ( G ) and f | M s : H q ( M s , M s − ) → H q ( M s , M s − ) are shift equivalentwhere ( M s ) ms = is a fixed filtration associated to the basic sets, ( W s ) ms = , of ( M , f ) . The reader might notice that Bowen’s and Franks’ result implies [1, Theorem 2](stated as Theorem 1 above). However, the proof in [2] uses [1, Theorem 2].
Example 10.
Two dimensional hyperbolic toral automorphisms
We give an example in which one can compute the standard homology, Putnam’shomology and the relevant actions explicitly. Let j = A = (cid:18) (cid:19) and consider the induced action on the two-torus, R / Z . In this example, W is theentire manifold and D is the constant function one and q = H N ( R / Z ; R ) ∼ = R : N = , R : N =
10 : otherwiseand the action is given by the identity on H ( R / Z ; R ) , A on H ( R / Z ; R ) , andminus the identity on H ( R / Z ; R ) .In regards to Putnam’s homology (based on [8, Example 7.4]) we have the fol-lowing H sN ( R / Z , A ) ⊗ R ∼ = R : N = − , R : N =
00 : otherwise signed version of Putnam’s homology theory: Lefschetz and zeta functions 13 and the action of (( j − ) s ) ⊗ Id R is given by the identity on H s − ( R / Z , A ) ⊗ R , A on H ( R / Z , A ) ⊗ R , and minus the identity on H s (( R / Z , A ) ⊗ R .Thus, in this very special case, there is an even stronger than predicted by The-orem 6 relationship between the homology of torus and Putnam’s homology of theSmale space ( R / Z , A ) . Namely, they are the same with dimension shift of one(this is exactly the rank of bundle E u in this case). Moreover, the actions induced by f and j − are also the same (again with dimension shift). Acknowledgements
I thank Magnus Goffeng, Ian Putnam and Robert Yuncken for discussions.In addition, I thank Magnus for encouraging me to publish these results. I also thank the refereefor a number of useful suggestions.
References
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