K-Theoretic Duality for Hyperbolic Dynamical Systems

K-THEORETI C DUALITY FOR HYPERB OLIC D YNAMICAL SYSTEMS JEROME KAMINKER, IAN F. P UTNAM 1 , AND MICHAEL F. WHITT AKER Abstract. The K-theoretic ana log of Spanier-Whitehead duality for no ncommutativ e C ∗ - algebras is sho wn to hold for th e Ruelle algebra s asso ciated to irr educible Smale spac e s. This had previously been prov ed only for shifts of ﬁnite t yp e. Implications of this result as well a s re la tions to the Ba um-Connes conjecture and other topics are also considered. 1. Intr oduction The goal of t his pap er is t o exhibit a dualit y b etw e en t w o C ∗ -algebras asso ciated to a h yp erb olic dynamical system. This is a noncomm utativ e v ersion of Spanier-Whitehead dualit y from to p ology . It turns out that it is a sp ecial case of a t yp e of dualit y whic h o ccurs in sev eral diﬀerent settings. It will b e describ ed carefully and we will indicate some of the diﬀeren t conte xts in whic h it app ears. Let us ﬁrst brieﬂy recall Spanier-Whitehead dualit y , a g eneralization of Alexander dualit y that relates the ho mology of a subspace of a sphere with the cohomolo g y of its complemen t. Giv en a ﬁnite complex, X ⊆ S n +1 , consider the map (1) ∆ : X × ( S n +1 \ X ) → S n , deﬁned b y ∆( x, y ) = x − y k x − y k , where the algebraic op eratio ns tak e place in S n +1 \{ north p ole } ∼ = R n +1 . Then one has an isomorphism (2) ∆ ∗ ([ S n ]) / : ˜ H n − i ( X ) → H i ( S n +1 \ X ) , giv en b y slant pro duct. This was generalized by Spanier and Whitehead to allo w S n +1 \ X to b e replaced b y a space of the ho mo t op y t yp e of a ﬁnite complex, D n ( X ), for whic h the analog o f the ab ov e relations hold. It is in this form that these notions extend natura lly to the noncomm utativ e setting. In noncomm utativ e topolog y , the roles of homolo gy and cohomology are pla y ed by K- theory and K- ho mology . These hav e b een com bined into a biv arian t theory by Kasparov, for whic h one has K-theory , K ∗ ( A ) = K K ∗ ( C , A ), and K-homolo gy , K ∗ ( A ) = K K ∗ ( A, C ). Kasparo v’s theory comes equipp ed with a product tha t is the analo g of the slant pro duct used ab ov e. Of course, noncomm utativ e C ∗ -algebras play the role of algebras of con tin uous functions on ordinary spaces. With this in hand, the g eneralization of Spanier-Whitehead dualit y to this setting is easily accomplished. Let A b e a C ∗ -algebra. W e will consider indices in K-theory to b e mo dulo 2. A C ∗ - algebra D n ( A ) is a Spanier-Whitehead n-dual of A , (or simply a “dual“ when the conte xt 2000 Mathematics Subje ct Classiﬁc ation. P rimary4 6L80;Seco ndary37B10,37D40 ,46L 80,46L 35. 1. Research suppo rted in part b y an NSER C Disco very Grant. 1 2 JER OME KAMINKER , IAN F. PUTNAM 1 , AND MICHAEL F. WH ITT AKER is clear) if there are duality classes δ ∈ K n ( A ⊗ D n ( A )) and ∆ ∈ K n ( A ⊗ D n ( A )) suc h that the Kasparov pro ducts yields inv e rse isomorphisms, (3) δ ⊗ D n ( A ) : K i ( D n ( A )) → K n − i ( A ) ⊗ A ∆ : K n − i ( A ) → K i ( D n ( A )) . Note that in the noncomm u tativ e case, giv en δ , it is an additional condition to require the existence of ∆, while in the comm utativ e case this holds automatically [41]. It is natural to compare this to the noncomm utativ e v ersion of K- theoretic P oincar ´ e du- alit y , a s in tro duced b y Connes. In general, these notions are diﬀeren t. The main thing is tha t P oincar ´ e duality relates K-theory and K-homology of the same algebra. Ho w ev e r, this is more restrictiv e than simply ﬁnding a dual algebra, B , whose K-theory is isomor- phic to the K-homology o f A . Ev en in cases where one can choose an a lgebra to b e it s o wn Spanier-Whitehead dual, care m ust b e take n. A situation whic h indicates this is when the a lgebra is A = C ( S 1 ). Since S 3 \ S 1 deformation retra cts to S 1 , w e may take C ( S 1 ) itself to b e a Spanier-Whitehead 2- dual. The dualit y class ∆ lies in K 0 ( C ( S 1 ) ⊗ C ( S 1 )) and the isomorphism from (3) pro vides isomorphisms K ∗ ( C ( S 1 )) ∼ = K ∗ ( C ( S 1 )). How eve r, S 1 is an o dd-dimensional S pin c manifold, so it has a K-theory f undamental class whic h can b e view ed as being in K 1 ( C ( S 1 ) ⊗ C ( S 1 )) and it pro vides the usual P oincar ´ e duality isomorphisms K ∗ ( C ( S 1 )) ∼ = K ∗ +1 ( C ( S 1 )). The main result of this pap er exhibits this duality for the stable a nd unstable Ruelle algebras associated to a Smale space, (i.e. a compact space X with a hyperb olic homeo- morphism, ϕ ). The stable and unstable sets for ϕ prov ide X with the structure of foliated space, in the sense o f Mo ore-Schoche t, in tw o diﬀeren t w a ys, whic h are transve rse to eac h other. The easiest example to visualize is the 2-dimensional tor us, T 2 with the hyperb olic toral auto morphism giv en by the matrix ϕ = [ 1 1 1 0 ]. The stable and unstable foliated space structures are the Kronec k er ﬂo ws for ang les θ and θ ′ , resp ectiv ely , where θ = tan − 1 ( − γ ), θ ′ = ta n − 1 ( γ − 1 ), a nd γ is the golden mean γ = (1 + √ 5) / 2. One ma y no w asso ciate to these structures their Connes foliation algebras, whic h, in this example, are isomorphic to A θ ⊗ K and A θ ′ ⊗ K , r esp ective ly , where A θ is the irrational rotation algebra. Thes e al- gebras are in teres ting in v ariants of the dynamics, and in the presen t case they happ en to b e isomorphic, although that is far from true in general. The y are simple algebras with a canonical (semi-ﬁnite) trace. The homeomorphism, ϕ , induces an automorphism of eac h of these algebras, and one can tak e the asso ciated crossed pro duct algebras, ( A θ ⊗ K ) ⋊ ϕ ∗ Z and ( A θ ′ ⊗ K ) ⋊ ϕ ∗ Z . One obtains in this w a y simple, purely , inﬁnite C ∗ -algebras. They are sp ecial cases of algebras in tro duced by the second author in [32] and are called the stable and unstable Ruelle alg ebras asso ciated to a Smale space. As a consequence o f Elliott’s classiﬁ- cation program, they ha v e the remark able prop ert y of b eing determined up to isomorphism b y their K -theory groups, [28]. It is these algebras whic h will b e shown to b e duals. W e brieﬂy review the general dynamical setting for the dualit y . T he necessary precise deﬁnitions will b e presen ted in Section 2. Let ( X, ϕ ) be a Smale space. In later sections, we will imp ose some mild dynamical conditions, but they will b e suppress ed in the in tro duc- tion. Consider the group oids give n by the equiv alence relatio ns of b eing stably o r unstably equiv alen t. In general, these a re analogs of the holonomy group oid of a foliation. They are lo cally compact g roup oids whic h a dmit Haar systems, so that one may deﬁne their C ∗ K-THEORETIC DU ALITY FOR HYPERBOLIC DYNAMICAL SY STEMS 3 -algebras, S ( X , ϕ ) and U ( X , ϕ ). These are ﬁnite direct sums of simple C ∗ -algebras whic h are also separable, n uclear, and stable. They are often known to be of the t yp e to which Elliott’s classiﬁcation pro gram (in the ﬁnite case) applies. They eac h hav e a densely deﬁned trace and the map ϕ induce s auto morphisms, α s , α u , on the algebras, whic h scale the trace b y the lo g arithm of the en trop y of ϕ . As ab ov e , w e now take the crossed pro ducts by these automorphisms to o btain the stable and unstable Ruelle algebras, R s ( X , ϕ ) = S ( X , ϕ ) ⋊ α s Z , R u ( X , ϕ ) = U ( X , ϕ ) ⋊ α u Z . These C ∗ -algebras are se parable, simple, stable, n uclear, purely inﬁnite, and satisfy the Uni- v ersal Co eﬃcien t Theorem. Th us, according to the purely inﬁnite case of Elliott’s program, as dev elop ed b y Kirc h berg and Phillips, they are completely classiﬁed by their K -theory groups. It is intere sting that these alg ebras, whic h arose from dynamics and dualit y theory , turn out to hav e remark able prop erties as C ∗ -algebras. The dualit y theorem sho ws that the stable and unstable Ruelle algebras are Spanier- Whitehead duals of eac h other. This implies that the K -theory of R s ( X , ϕ ) is isomorphic, with a dimension shift, to the K -homology of R u ( X , ϕ ), and vice-v ersa. 1.1. Theorem. L et ( X, ϕ ) b e an irr e ducible S male Sp ac e. Ther e exi s ts duality cla sses δ ∈ K K 1 ( C , R s ( X , ϕ ) ⊗ R u ( X , ϕ ) and ∆ ∈ K K 1 ( R s ( X , ϕ ) ⊗ R u ( X , ϕ ) , C ) such that δ ⊗ R u ( X,ϕ ) ∆ ∼ = 1 R s ( X,ϕ ) and δ ⊗ R s ( X,ϕ ) ∆ ∼ = − 1 R u ( X,ϕ ) . T aking the Kasparov pro duct with δ and ∆ yields inv erse isomorphisms K ∗ ( R s ( X , ϕ ) ∆ ⊗ R s ( X,ϕ ) ✲ K ∗ +1 ( R u ( X , ϕ )) K ∗ ( R u ( X , ϕ )) δ ⊗ R u ( X,ϕ ) ✲ K ∗ +1 ( R s ( X , ϕ )) . W e will now describ e some of the ba c kground for our results. In [31], the second a ut ho r in tro duced the a lg ebras studied in the presen t pap er. They w ere based on constructions of algebras due t o Ruelle in [39 ], a lt ho ugh the crossed pro duct by the a utomorphism induced b y ϕ w as not used there. F urt her prop erties of thes e algebras w ere presen ted in [34]. In [21], the ﬁrst t w o authors work ed out a sp ecial case of t he duality theory when the Smale space was a subshift of ﬁnite t yp e. In this case, the Ruelle algebras we re kno wn to b e isomorphic to stabilized Cun tz- K rieger algebras. Sp eciﬁcally , supp ose that the Smale space w as the shift of ﬁnite t yp e asso ciated with a non- negativ e in tegral, irreducible matrix, A . Denote the dynamical system by ( X A , ϕ A ). It follow s that R s ( X A , ϕ A ) ∼ = O A t ⊗ K and R u ( X A , ϕ A ) ∼ = O A ⊗ K . F ollo wing w ork of D. Ev ans [14] and D. V oiculescu [45], one considers the full F o ck space o f a ﬁnite dimensional Hilb ert space a nd the asso ciated creation and annihilation op erators. One compresses them to a subspace determined b y the matrix A and generates a C ∗ -algebra, E . This algebra con tains the compact op erators and one o btains an extension, 0 ✲ K ✲ E ✲ O A ⊗ O A t ✲ 0 . 4 JER OME KAMINKER , IAN F. PUTNAM 1 , AND MICHAEL F. WH ITT AKER The extension determines a n elemen t ∆ in K 1 ( O A ⊗ O A t ) and it w a s sho wn in [21] that it induces the required dualit y isomorphism. J. Zac harias and I. P op escu [3 0] hav e extended this t yp e of dualit y theory to higher ra nk graph algebras. There are other sources for examples of dualit y . One t yp e is based on an amenable action of a hyperb o lic group, Γ, on a compact space, X . In this setting, the crossed pro duct algebra, C ( X ) ⋊ Γ can often b e shown to b e it s o wn dual, as in Poinc ar´ e dualit y . Since the dualit y presen ted in this pap er is based on transv ersalit y coming from hyperb olic dynamics, it is natural to ask wh y this o ccurs. The underlying idea is that the action of Γ can b e reco ded so it ha s the same orbit structure a s a single hy p erb olic transformation. Using this principle and t he results in [4], J. Spielb erg show ed that for certain F uchs ian gro ups acting on their b oundary , the crossed pro duct algebra w as isomorphic to a Cun tz-Krieger algebra O A . This was extended b y La ca and Spielb erg, [23], (see also C. Anan t ha raman-Delaro c he [1]) to show that the crossed pro duct, C ( S 1 ) ⋊ S L (2 , Z ) is isomorphic to a Ruelle algebra. One may ask how general a phenomenon t his is. According to Connes-F eldman-W eiss, [7], an amenable action is orbit equiv alent, in the measure theoretic con text, t o t he action of a single tr a nsformation. In the cases at hand, one gets m uc h more than a orbit equiv alence in the measure category , a nd the ma p it induces provide s isomorphisms of the crossed pro duct algebras with Ruelle algebras of asso ciated hy p erb olic dynamic al systems. It w ould b e in teresting to ha v e a theory in termediate betw een measure theoretic orbit equiv alence, whic h is nat ur a lly asso ciated to von Neumann algebras, and top o logical ﬂow equiv alence, whic h is related to C ∗ -algebras, where an equiv alence w ould induce isomorphisms of Ruelle ty p e algebras. This line w a s pursued further by H. Emerson, who, without assuming that suc h crossed pro ducts are related to hyperb o lic dynamical systems, w a s able to sho w that if one take s Γ to b e a Gromov hy p erb olic group, then there is an extension, analogous to the one ab o v e, whose K -ho mo lo gy class yields an isomorphism K ∗ ( C ( ∂ Γ) × Γ) ∼ = K ∗ +1 ( C ( ∂ Γ) × Γ) [13 ]. W e will describe in Section 4 how this clariﬁes a relation b et w een the dynamical dualit y of the presen t pap er and the Baum-Connes map for hyperb o lic gro ups. Building on ideas lik e these, the dualit y theory for general Smale spaces ha s play ed a role in the w ork of V. Nekrashevy c h [2 6] on pro viding more precision to Sulliv an’s dictionary relating the dynamics of rational maps to that of Kleinian groups. Starting with a rational map, f : C → C , suitably r estricted, Nekrashevyc h constructs a self-similar g r oup Γ f , the iterated monodro m y group of f . This group ha s a limit set, Λ(Γ f ), whic h admits a self map, λ f , so that the pair (Λ(Γ f ) , λ f ) is top ologically conjugate to ( f , Julia( f )), the latter b eing f restricted to it s Julia set. He sho ws that t he in v erse limit lim ← − { Λ(Γ f ) , λ f } is a Smale space. Th us, one ma y study the stable and unstable Ruelle algebras and their dualit y in this con text. Nekrashev yc h establishes that the unstable Ruelle algebra is Morita equiv alen t t o an algebra asso ciated to the iterated mono drom y group, while the stable Ruelle algebra is Morita equiv a lent to the D eacon u- Renault algebra [1 1, 36] asso ciated to the rationa l map, f . One ma y thus view the dynamical duality b et w een the Ruelle algebras as relating the expanding dynamics of f with the contracting dynamics asso ciated to the action o f the self- similar group on its limit set. T o get closer to Sulliv a n’s program, it would remain to relate the latter with the action of a Kleinian group on its limit set. If the resp ective algebras a r e K-THEORETIC DU ALITY FOR HYPERBOLIC DYNAMICAL SY STEMS 5 simple and purely inﬁnite, as exp ected, then ﬁnding a Kleinian group whose algebras hav e the same K-theory a s t he self-similar group would pro vide supp ort for Sulliv an’s dictiona r y . The structure of the pap er is as follow s. Section 2 is an in tro duction to Smale spaces. W e pro vide here mo r e details on the techn ical asp ects needed fo r later pro o fs. In Sections 3 , w e review the construction of the C ∗ -algebras asso ciat ed to Smale spaces. Section 4 cov ers the K -theoretic duality we will use a nd describ es v arious con texts where it arises naturally . In Section 5, the K-theory duality elemen t is constructed. This is essen tially a consequence of the transv ersality of the stable and unstable equiv alence relations, and can b e constructed in more generality . Indeed, the Mishc henk o line bundle, used in the Baum-Connes assem bly map, can b e obtained this w ay . Se ction 6 is dev oted to construction of t he K-homology dualit y class, whic h prov ides the inv erse on K-theory . Here, the hyperb o lic nature of t he dynamics pla ys a crucial role. Th us, it app ears that t he K-t heory dualit y class exists in reasonable generalit y , but the existence of an inv erse requires additional structure. This is precisely analogous to the diﬃcult y in ﬁnding an inv erse to the Baum-Connes a ssem bly map using the Dirac-dual Dirac method. In Section 7, the pro of of the main theorem will b e completed and, ﬁnally , in Section 8 w e will discuss op en questions and p ossible extensions of the theory . 2. Smale sp aces In this section, w e pro vide a brief in tro duction to Smale spaces. The reader is also referred to [31, 38], but w e will try to kee p our treatmen t self-contained. W e a ssume that ( X , d ) is a compact metric space and that ϕ is a homeomorphism. The main gist of the deﬁnition is tha t, lo cally at a p oin t x , X is homeomorphic to the product of t wo subsets, denoted X s ( x, ε ) and X u ( x, ε ) and on these, the maps ϕ and ϕ − 1 , resp ectiv ely , are contracting. F or the uninitiated, it ma y b e b est to b egin reading from Figure 1 to Deﬁnition 2.2 a nd then work backw ards t o the more rig orous asp ects b elo w. W e assume the existence of constan t s, ε X > 0 , λ > 1 , and a map ( x, y ) ∈ X, d ( x, y ) ≤ ε X 7→ [ x, y ] ∈ X satisfying a num b er of conditions. F irst, [ , ] is join tly contin uo us o n its domain of deﬁnition. Also, it satisﬁes [ x, x ] = x, [ x, [ y , z ]] = [ x, z ] , [[ x, y ] , z ] = [ x, z ] , ϕ [ x, y ] = [ ϕ ( x ) , ϕ ( y )] , for an y x, y , z in X , where b oth sides of the equalit y a re deﬁned. It follow s easily from t hese axioms that [ x, y ] = x if and o nly if [ y , x ] = y and [ x, y ] = y if and only if [ y , x ] = x . W e deﬁne, for each x in X and 0 < ε ≤ ε X , sets X s ( x, ε ) = { y ∈ X | d ( x, y ) ≤ ε, [ y , x ] = x } , X u ( x, ε ) = { y ∈ X | d ( x, y ) ≤ ε, [ x, y ] = x } . It follows easily from the axioms tha t the map [ , ] : X u ( x, ε ) × X s ( x, ε ) → X 6 JER OME KAMINKER , IAN F. PUTNAM 1 , AND MICHAEL F. WH ITT AKER is a homeomorphism to its image, whic h is a neigh b ourho o d of x in X , pro vided ε ≤ ε X / 2. The in v erse map sends a p o int z close to x to the pair ([ z , x ] , [ x, z ]) . Moreov er, a s w e v ary ε , the images form a neigh b ourho o d ba se for the top o logy at x . T o summarize , X has a lo cal pro duct structure. W e giv e a pro o f of the following simple result for completeness a nd b ecause we will use it later in an essen tia l wa y . 2.1. Lemma. Given two p oints x, y in X and 0 < ε ≤ ε X / 2 , if the interse ction X s ( x, ε ) ∩ X u ( y , ε ) is non-emp ty, then it is the single p o i n t [ x, y ] . Pr o o f . Supp ose that z is in the in tersection. This means that [ z , x ] = x a nd [ y , z ] = y . It also means that d ( x, y ) ≤ d ( x, z ) + d ( z , y ) < 2 ε ≤ ε X so that [ x, y ] is deﬁned. It follo ws that [ x, y ] = [[ z , x ] , [ y , z ]] = [[ z , x ] , z ] = [ z , z ] = z .  It will probably help to hav e a picture of the bra c ke t in mind, see ﬁgure 1. X s ( x, ε X ) X u ( x, ε X ) x [ x, y ] X s ( y , ε X ) X u ( y , ε X ) y [ y , x ] Figure 1. The bra ck et map The ﬁnal axiom is that, for y , z in X s ( x, ε X ), w e hav e d ( ϕ ( y ) , ϕ ( z )) ≤ λ − 1 d ( y , z ) , and for y , z in X u ( x, ε X ), w e hav e d ( ϕ − 1 ( y ) , ϕ − 1 ( z )) ≤ λ − 1 d ( y , z ) . That is, on the set X s ( x, ε X ), ϕ is con tracting. It is tempting to sa y that on X u ( x, ε X ), ϕ is expanding, but it is b etter to say that its in verse is con tra cting. 2.2. D eﬁnition. A Smale space is a compact metric space ( X , d ) with a homeomorphism ϕ suc h that there exist constants ε X > 0 , λ > 1 and map [ , ] satisfying the conditions a b o v e. A Smale space ( X , d, ϕ ) is said to b e irreducible if the set of p erio dic p oin ts under ϕ are dense and there is a dense ϕ -orbit. The sets X s ( x, ε ) a nd X u ( x, ε ) a r e called the lo cal con tracting and expanding sets, res p ec- tiv ely . W e note for later con v enience, that if y is in X s ( x, ε ), f or some x, y , ε , then for all k ≥ 0, ϕ k ( y ) is in X s ( ϕ k ( x ) , λ − k ε ). Similarly , if y is in X u ( x, ε ), for some x, y , ε , then for all k ≥ 0, ϕ − k ( y ) is in X u ( ϕ − k ( x ) , λ − k ε ). K-THEORETIC DU ALITY FOR HYPERBOLIC DYNAMICAL SY STEMS 7 2.3. Lemma. Th e r e is a c onstant 0 < ε ′ X ≤ ε X / 2 such that, if d ( x, y ) < ε ′ X , then b oth d ( x, [ x, y ]) , d ( y , [ x, y ]) < ε X / 2 and hen c e [ x, y ] is in X s ( x, ε X / 2) and in X u ( y , ε X / 2) . Pr o o f . The functions d ( x, [ x, y ]) , d ( y , [ x, y ]) are b oth deﬁned o n the set of pa irs ( x, y ) with d ( x, y ) ≤ ε X , whic h is compact, and are con tinuous . Moreov er, on the set where x = y , they hav e v alue zero. The existence of ε ′ X satisfying the ﬁrst conditions follows from uniform con tinuit y . The last pa r t follo ws from t he deﬁnitions.  W e no w deﬁne global stable and unstable equiv alence relations o n X . Giv en a p oint x in X we deﬁne the stable a nd unstable equiv alence classes of x by X s ( x ) = { y ∈ X | lim n → + ∞ d ( ϕ n ( x ) , ϕ n ( y )) = 0 } , X u ( x ) = { y ∈ X | lim n → + ∞ d ( ϕ − n ( x ) , ϕ − n ( y )) = 0 } . W e will often denote stable equiv alence b y x ∼ s x ′ and unstable equiv alence b y y ∼ u y ′ . T o see the connection b et w een these g lo bal stable and unstable sets, w e no t e that, for any x in X and ε > 0, X s ( x, ε ) ⊂ X s ( x ) , X u ( x, ε ) ⊂ X u ( x ). Moreov er, y in X is in X s ( x ) (or X u ( x )) if and only if there exists n ≥ 0 suc h that ϕ n ( y ) is in X s ( ϕ n ( x ) , ε ) (or ϕ − n ( y ) is in X u ( ϕ − n ( x ) , ε ), resp ective ly). 3. C ∗ -algebras W e describe the construction o f C ∗ -algebras from a Smale space. In the introduction, w e indicated that the C ∗ -algebras S ( X , ϕ ) a nd U ( X , ϕ ) are t he C ∗ -algebras of the stable and unstable equiv alence relations, resp ectiv ely . This is correct in spirit, but for the purp oses o f this pap er, it is a half-truth. W e will ﬁnd it muc h easier to work with equiv alence relations whic h are equiv alen t t o these (in the sense of Muhly , Renault and Williams [2 5]) but whic h are ´ etale. T o do this, we simply restrict to the stable and unstable equiv alence classes of ϕ - in v ariant sets of p oints. Some care m ust b e ta ken b ecause these unstable class es a r e endo we d with a diﬀeren t (and more natural top o logy) t han the relative top olog y of X . Sp eciﬁcally , w e c ho ose se ts, P and Q , consisting of p erio dic p oints and their or bits. W e note that, at this p o in t, there are no limitations on the sets P and Q , ho wev er, later w e require that they a re distinct from one another. W e shall then construct ´ etale group oids of stable and unstable equiv alence, G s ( X , ϕ, Q ) and G u ( X , ϕ, P ). The C ∗ -algebras of these gro up oids will b e denoted S ( X , ϕ, P ) and U ( X , ϕ, P ), resp ectiv ely . Let ( X , d , ϕ ) b e a Smale space and let P a nd Q b e ﬁnite sets of ϕ -inv ariant p erio dic p oints. Consider X s ( P ) = [ p ∈ P X s ( p ) , X u ( Q ) = [ q ∈ Q X u ( q ) . The set X s ( P ) is endow ed with lo cally compact and Hausdorﬀ top ology b y declaring that the collection of sets X s ( x, ε ), as x v ar ies ov er X s ( P ) and 0 < ε < ε X , forms a neighbourho o d base. Similarly for X u ( Q ). The stable and unstable group oids are then deﬁned by G s ( X , ϕ, Q ) = { ( v , w ) | v ∼ s w and v , w ∈ X u ( Q ) } G u ( X , ϕ, P ) = { ( v , w ) | v ∼ u w and v , w ∈ X s ( P ) } . 8 JER OME KAMINKER , IAN F. PUTNAM 1 , AND MICHAEL F. WH ITT AKER Let ( v , w ) b e in G s ( X , ϕ, Q ). Then for some N ≥ 0, ϕ N ( w ) is in X s ( ϕ N ( v ) , ε ′ X ). By the con tinuit y of ϕ , w e may ﬁnd δ > 0 suc h that ϕ N ( X u ( w , δ )) ⊂ X u ( ϕ N ( w ) , ε ′ X ). A map h s : X u ( w , δ ) → X u ( v , ε X / 2) is deﬁned b y h s ( x ) = ϕ − N [ ϕ N ( x ) , ϕ N ( v )] . Moreo ver, it is the comp osition of t hree maps, ϕ N , [ · , ϕ N ( v )], and ϕ − N , which are op en on lo cal unstable sets, and hence it is op en. It is easy to v erify tha t this map is a lo cal homeo- morphism and that interc hanging the roles o f v a nd w giv es ano t her lo cal homeomorphism. Where comp osition of the tw o maps is deﬁned it is the identit y , in either order. L et V s ( v , w, h s , δ ) = { ( h s ( x ) , x ) | x ∈ X u ( w , δ ) } . It will help t o ha v e a picture of the map h s : S U w x S U v h s ( x ) = φ − N [ φ N ( x ) , φ N ( v )] S U U φ N ( w ) φ N ( x ) φ N ( v ) [ φ N ( x ) , φ N ( v )] φ N φ − N Figure 2. The lo cal homeomorphism h s : X u ( w , δ ) → X u ( v , δ ) 3.1. Lemma. The c ol le ction of sets V s ( v , w, h s , δ ) as ab ove forms a neighb o urho o d b ase for a top olo gy on G s ( X , ϕ, Q ) in which it is an ´ etale gr oup oid. W e also remark that if Q meets eac h irreducible comp onen t of ( X , d, ϕ ), then this gr o up oid is unique up to the notion of equiv alence giv en in [25]. In [34], it is sho wn that these group oids are amenable. Analogous results obvious ly hold for G u ( X , ϕ, P ). K-THEORETIC DU ALITY FOR HYPERBOLIC DYNAMICAL SY STEMS 9 Let C c ( G s ( X , ϕ, Q )) denote the contin uous functions of compact supp ort on G s ( X , ϕ, Q ), whic h is a complex linear space. A pro duct a nd inv o lution a r e deﬁned o n C c ( G s ( X , ϕ, Q )) as follows , for f , g ∈ C c ( G s ( X , ϕ, Q )) and ( x, y ) ∈ G s ( X , ϕ, Q ), f · g ( x, y ) = X z ∼ s x f ( x, z ) g ( z , y ) f ∗ ( x, y ) = f ( y , x ) . This mak es C c ( G s ( X , ϕ, Q )) in to a complex ∗ -algebra. An y function in C c ( G s ( X , ϕ, Q )) ma y b e written as a sum of functions, eac h hav ing supp ort in an elemen t of the neighbourho o d base describ ed in 3.1. 3.2. Deﬁnition. Let ( X , d, ϕ ) b e a Smale space and P and Q b e ϕ -inv arian t sets of p erio dic p oin ts. W e deﬁne S ( X , ϕ, Q ) and U ( X , ϕ, P ) to b e the (reduced) C ∗ -algebras asso ciated with the ´ etale group oids G s ( X , ϕ, Q ) and G u ( X , ϕ, P ), resp ectiv ely . When no confusion will arise, w e denote t hem b y S and U . W e now w an t to deﬁne a canonical represen ta tion of these C ∗ -algebras. Let X h ( P , Q ) denote the set X s ( P ) ∩ X u ( Q ). Since ( X, d , ϕ ) is assumed to b e irreducible, X h ( P , Q ) is dense in X [38 ]. Viewing X h ( P , Q ) as the set of G s ( X , ϕ, Q ) equiv alence classe s of P , w e consider the restriction of the regular represen tation of S ( X , ϕ, Q ) to equiv alence classes of P in G s ( X , ϕ, Q ). This means that w e consider the Hilb ert space H = ℓ 2 ( X h ( P , Q )), a nd for a in C c ( G s ( X , ϕ, Q )), ξ in H , w e deﬁne ( aξ )( x ) = X ( x,y ) ∈ G s ( X,ϕ,Q ) a ( x, y ) ξ ( y ) ( x in H ) . Notice that w e suppress the notation for the r epresen tation of C c ( G s ( X , ϕ, Q )) as b ounded op erators on B ( H ). Next, viewing the same set X h ( P , Q ) as the union of G u ( X , ϕ, P )-equiv alence classes of p oin ts in Q , w e can consider the the restriction of the regular r epresen tation of U ( X , ϕ, P ) to the same Hilb ert space H . F o r b in C c ( G u ( X , ϕ, P )), ξ in H , we deﬁne ( bξ )( x ) = X ( x,y ) ∈ G u ( X,ϕ,P ) b ( x, y ) ξ ( y ) ( x in H ) . F o r each x in X h ( P , Q ), w e let δ x denote the function on X h ( P , Q ) taking v alue 1 at x and zero elsewhere. Of course, the set { δ x | x ∈ X h ( P , Q ) } forms a ba sis for the Hilb ert space H = ℓ 2 ( X h ( P , Q )). The following t wo lemmas follow directly fro m the deﬁnitions. W e omit the pro o f s. 3.3. Lemma. L et V s ( v , w, N , δ ) b e a b asic op en set in G s ( X , ϕ, Q ) and supp ose a is a c on- tinuous c omp actly supp orte d function on V s ( v , w, h s , δ ) . F o r e ac h x in X h ( P , Q ) , we have aδ x = a ( h s ( x ) , x ) δ h s ( x ) , wher e the right ha nd s ide is deﬁne d to b e zer o if h s ( x ) is not deﬁne d. Deﬁne S our ce ( a ) ⊆ X u ( w , δ ) to b e the p oints for whic h a is non-zer o on its domain and deﬁne Rang e ( a ) ⊆ X u ( v , ε ′ X ) to b e the p oints in X u ( v , ε ′ X ) for which a ( h s ( x ) , x ) δ h u ( x ) is non-z e r o . Observe that a is zer o on the ortho gona l c o m plement of X u ( w , δ ) . 10 JER OME KAMINKER , IAN F. PUTNAM 1 , AND MICHAEL F. WH ITT AKER 3.4. Lemma. L et V u ( v , w, N , δ ) b e a b asic op en set in G u ( X , ϕ, P ) an d supp ose b is a c on- tinuous c omp actly supp orte d function on V u ( v , w, h u , δ ) . F or e ach x in X h ( P , Q ) , we have bδ x = a ( h u ( x ) , x ) δ h u ( x ) , wher e the right hand side is deﬁne d to b e zer o if h u ( x ) is not deﬁne d. D eﬁne S our ce ( b ) ⊆ X s ( w , δ ) to b e the p oints for which b is non-zer o on its domain and deﬁne Rang e ( b ) ⊆ X s ( v , ε ′ X ) to b e the p oints in X s ( v , ε ′ X ) for w h ich b ( h u ( x ) , x ) δ h s ( x ) is non-zer o. Obs e rve that b is zer o on the ortho gonal c om plement of X s ( w , δ ) . W e note that ev ery elemen t of either of the ab ov e C ∗ -algebras can b e uniformly appro xi- mated b y a ﬁnite sum of functions supp orted in a neigh b ourho o d base sets. W e also hav e a nice geometric picture of the Hilb ert space H in the sense that a na t ura l basis is parameter- ized b y the p oints of X h ( P , Q ). In this spirit, the picture o f the map h s , giv en in Figure 2 on pag e 8, can no w b e view ed as a picture of the op erator a , up to the v a lue of the function at sp eciﬁc p oin ts. The sets of p erio dic p oints P and Q are c hosen t o be ϕ - inv ariant a nd so X s ( P ) and X u ( Q ) are also. Moreov er, it is clear that ϕ induces homeomorphisms o f b oth these spaces. It is also clear that ϕ × ϕ induces automor phisms of G s ( X , ϕ, Q ) and G u ( X , ϕ, P ). These in turn deﬁne automorphisms of the C ∗ -algebras, S ( X , ϕ, Q ) and U ( X , ϕ, P ), denoted α s and α u , resp ectiv ely . Sp eciﬁcally , suppo se a is a con tin uo us compactly supp orted function on a ba sic set V s ( v , w, h s , δ ) and x is in X h ( P , Q ), then w e ha v e α s ( a ) δ x = a ( h s ◦ ϕ − 1 ( x ) , ϕ − 1 ( x )) δ ϕ ◦ h s ◦ ϕ − 1 ( x ) . Similarly , if b is a con tin uous compactly suppo rted function o n a basic set V u ( v , w, h u , δ ) and x is in X h ( P , Q ), then we ha v e α u ( b ) δ x = b ( h u ◦ ϕ − 1 ( x ) , ϕ − 1 ( x )) δ ϕ ◦ h u ◦ ϕ − 1 ( x ) . F o r the same reason X h ( P , Q ) is also ϕ -inv ariant and this implies that there is a canonical unitary op erat o r u on H deﬁned by uξ = ξ ◦ ϕ − 1 , for an y ξ in H . W e note that uδ x = δ ϕ ( x ) , for an y x in X h ( P , Q ). W e a lso note, without pro of, that uau ∗ = α s ( a ) , a ∈ S ( X , ϕ , Q ) ubu ∗ = α u ( b ) , b ∈ U ( X , ϕ , P ) . These co v aria n t pairs deﬁne crossed pro duct C ∗ -algebras. 3.5. Deﬁnition. The stable and unstable Ruelle algebras, denoted b y R S ( X , ϕ, Q ) and R U ( X , ϕ, P ), resp ectiv ely , are the crossed pro ducts: R S ( X , ϕ, Q ) = S ( X , ϕ, Q ) ⋊ α s Z and R U ( X , ϕ, P ) = U ( X , ϕ, P ) ⋊ α u Z . W e remark that the Ruelle algebras, as deﬁned here, ar e Morita equiv alent to the Ruelle algebras deﬁned in [31]. Moreov er, the Ruelle algebras we re sho wn to b e separable, simple, stable, nucle ar, and purely inﬁnite when ( X , d, ϕ ) is irreducible [3 4], hence they also satisfy the Univ ersal Coeﬃcien t Theorem [44]. Moreov er, according to t he purely inﬁnite case of Elliott’s classiﬁcation pro gram, as dev elop ed b y Kirch b erg and Phillips, they are completely classiﬁed by their K -theory gro ups. Due to the deﬁnitions of stable a nd unstable equiv alence w e also note that t ha t U ( X , ϕ, Q ) = S ( X , ϕ − 1 , Q ) and R U ( X , ϕ, Q ) = R S ( X , ϕ − 1 , Q ). K-THEORETIC DU ALITY FOR HYPERBOLIC DYNAMICAL SY STEMS 11 4. Noncommut a tive D uality In this section w e will review ba sic facts ab out dua lity whic h w e will need. W e will also desc rib e some prop erties p ossesse d by algebras whic h ha v e duals and discus s v arious examples. Muc h o f this materia l can b e f ound in [2 1] or [13]. 4.1. KK-theory. Let K K 0 ( A, B ) denote the Ka sparo v K K -g r o up fo r a pair of separable C ∗ -algebras A and B . Denote C 0 (0 , 1) by S . Then K K 1 ( A, B ) = K K 0 ( A ⊗ S , B ). There are v arious w a ys of obtaining elemen ts of K K 0 ( A, B ). F or ex ample, a ny homomorphism, h : A → B determines an elemen t [ h ] ∈ K K 0 ( A, B ). If A is nucle ar then there is a natural isomorphism K K 1 ( A, B ) ∼ = E xt ( A, B ) [20], where E xt ( A, B ) is the gr o up of classes of C ∗ - algebra extensions of the form (4) 0 → B ⊗ K → E → A → 0 . Th us, an ex tension determines a n elemen t of K K 1 ( A, B ). O ne can retriev e the ordinary K-theory and K- homology groups fro m KK -theory as K ∗ ( A ) = K K ∗ ( C , A ) (5) K ∗ ( A ) = K K ∗ ( A, C ) . (6) W e will b e using the Kasparov pro duct, (7) K K i ( A, B ⊗ D ) × K K j ( D ⊗ A ′ , B ′ ) ⊗ D ✲ K K i + j ( A ⊗ A ′ , B ⊗ B ′ ) . As usual, indices ar e to b e take n mo dulo 2. In the course o f pro ofs it will b e necessary to b e more precise ab out explicit expressions for pro ducts. W e refer to Connes’ b o ok, [6], p. 42 8 , for a presen tation of this material, and [2 ] for a more complete t r eatmen t. Let 1 D ∈ K K 0 ( D , D ) denote the class determined b y the identit y homomorphism. Then there are natural maps τ D : K K i ( A, B ) → K K i ( A ⊗ D , B ⊗ D ) and τ D : K K i ( A, B ) → K K i ( D ⊗ A, D ⊗ B ) obtained via x 7→ x ⊗ 1 D and x 7→ 1 D ⊗ x . 4.2. Dualit y classes. 4.1. D eﬁnition. Let A and B b e separable C ∗ -algebras. W e sa y that A and B ar e Sp anier- Whitehe ad dual , or j ust dual , if there a r e duality classes ∆ ∈ K K i ( A ⊗ B , C ) and δ ∈ K K i ( C , A ⊗ B ) suc h that δ ⊗ B : K j ( B ) → K i + j ( A ) ⊗ A ∆ : K j ( A ) → K i + j ( B ) yield inv erse isomorphisms. The main criterion is the follow ing theorem, ﬁr st presen ted in Connes’ b o ok. (c.f. [6 , 21, 13]) 4.2. Theorem. L et ∆ ∈ K K i ( A ⊗ B , C ) and δ ∈ K K i ( C , A ⊗ B ) b e given, satisfying the two c onditions, δ ⊗ B ∆ = 1 A δ ⊗ A ∆ = ( − 1) i 1 B . 12 JER OME KAMINKER , IAN F. PUTNAM 1 , AND MICHAEL F. WH ITT AKER Then δ and ∆ imp l e ment a duality b etwe en A and B . Note that we are making use of the fo llowing standa r d conv entions to mak e sense of the form ulas in Theorem 4.2. Let σ : A ⊗ B → B ⊗ A b e the isomorphism interc hanging the factors. δ ⊗ B ∆ = σ ∗ ( δ ⊗ B σ ∗ (∆)) δ ⊗ A ∆ = σ ∗ ( σ ∗ ( δ ) ⊗ A ∆) . 4.3. Bott p erio dicit y and duality maps. Because of our con v ention in the deﬁnition of K K 1 ( A, B ), for the sequel w e will hav e to b e explicit a b out how Bott p erio dicity ﬁts in to this fo r the sequel. W e will also ha v e to b e more precise ab out the maps b et w een K -groups induced b y the dualit y elemen ts. Let T denote the T o eplitz extension 0 ✲ K ( ℓ 2 ( N )) ✲ T ✲ C ( S 1 ) ✲ 0 , whic h determin es an elemen t of K K 1 ( C ( S 1 ) , C ). Observ e that S ⊂ C ( T ) and we denote the restriction of the T o eplitz extension to S b y T 0 , whic h, b y our con v entions, is an ele men t of K K ( S ⊗ S , C ). No w if β ∈ K K ( C , S ⊗ S ) is the Bott elemen t, see 19.2.5 in [2], then w e hav e β ⊗ S ⊗ S T 0 = 1 C and T 0 ⊗ β = 1 S ⊗ S , (for this see Section 19 . 2 in [2]). That is, 1 A ∼ = τ A ( T 0 ) and 1 B ∼ = τ B ( T 0 ). In the presen t pap er w e will b e w orking only with o dd dualit y classes ∆ ∈ K K 1 ( A ⊗ B , C ) and δ ∈ K K 1 ( C , A ⊗ B ). W e obtain maps b et w een the v ario us K - groups asso ciated with A and B via t he Kasparo v pro duct and w e will need to b e more precise ab out their relation to Bott p erio dicit y . T o this end, let A and B b e C ∗ -algebras. Consider the homomorphisms ∆ i : K i ( A ) → K i +1 ( B ) and δ i : K i ( B ) → K i +1 ( A ) deﬁned b y ∆ 0 ( x ) = x ⊗ A ∆ x ∈ K 0 ( A ) , ∆ 1 ( x ) = β ⊗ S ⊗ S ( σ ∗ ( x ⊗ A ∆)) x ∈ K 1 ( A ) , δ 1 ( y ) = β ⊗ S ⊗ S ( δ ⊗ B y ) y ∈ K 1 ( B ) , δ 0 ( y ) = δ ⊗ B y y ∈ K 0 ( B ) . The comp ositions of these maps is described in the follow ing result from [13], whic h gen- eralizes one fr o m [21]. 4.3. Theorem ([13]) . L et A and B b e C ∗ -algebr as. S upp o s e the classes ∆ ∈ K K 1 ( A ⊗ B , C ) and δ ∈ K K 1 ( C , A ⊗ B ) satisfy the criterion in Th e or em 4.2. Then, δ i +1 ◦ ∆ i = ( − 1) i 1 K i ( B ) ∆ i +1 ◦ δ i = ( − 1) i +1 1 K i ( A ) In terchanging the roles of A and B g iv es a similar result, and in either case w e obtain isomorphisms K i ( B ) ∼ = K i +1 ( A ). K-THEORETIC DU ALITY FOR HYPERBOLIC DYNAMICAL SY STEMS 13 4.4. Consequences of duality . In this section w e will describe some algebraic consequenc es of an algebra ha ving a dual. Since this pap er deals with Ruelle algebras asso ciated to h yp er- b olic dynamical systems we will tak e adv antage of the additio na l prop erties these alg ebras ha ve. In particular, they are separable, nucle ar, purely inﬁnite, and since they are algebras obtained from amenable group oids, they satisfy the Univ ersal Co eﬃcien t Theorem, [37] . W e will sho w that the R uelle algebras are self-dual, hence satisfy a v ersion of Poincar ´ e duality . This requires an additional hy p othesis on the dynamical system which is v ery like ly to hold in general. Namely , w e a ssume that K ∗ ( U ( X , ϕ, P )) and K ∗ ( S ( X , ϕ , Q )) are ﬁnite rank ab elian groups. Indeed, it will follow f r o m results due to the second author on a sp ecial homology theory for Smale spaces [32], and is the analog of the rationality of the zeta function of suc h a system. T o start, w e will a ssume that A is separable and p ossess es the follo wing prop erties. a) The algebra A has an o dd, separable, Spanier-Whitehead dual, D ( A ). b) The Unive rsal Co eﬃcien t Theorem holds for A and D ( A ), with K-homolog y in the middle. It follows from this that, c) The Univers al Co eﬃcien t Theorem holds for A a nd D ( A ), with K-theory in the middle. d) K ∗ ( A ) and K ∗ ( A ) are ﬁnitely generated groups. Statemen t (c) follow s by applying duality to the Univ ersal Coeﬃen t Theorem for K- homology . W e will give a pro of of (d). Pr o o f . First note that the Univ ersal Co eﬃcien t Theorem and separability of A imply that Hom( K ∗ ( A ) , Z ) a nd Ext( K ∗ ( A ) , Z ) are b oth countable, as are the corresp onding groups with K ∗ ( A ) via dualit y . Let tK ∗ ( A ) denote the tor sion subgroup of K ∗ ( A ). Applying Hom( · , Z ) to the sequence 0 → tK ∗ ( A ) → K ∗ ( A ) → K ∗ ( A ) /tK ∗ ( A ) → 0 one deduces that Ext( K ∗ ( A ) /tK ∗ ( A ) , Z ) is countable. It is shown in [27] that if a gro up H is torsion free and Ext( H , Z ) is countable, then H is free. Applying this to K ∗ ( A ) /tK ∗ ( A ) one gets that K ∗ ( A ) = tK ∗ ( A ) ⊕ K ∗ ( A ) /tK ∗ ( A ). It fo llows, since Hom( K ∗ ( A ) , Z ) is coun table, that Hom( K ∗ ( A ) /tK ∗ ( A ) , Z ) is countable as w ell. Thu s, K ∗ ( A ) /tK ∗ ( A ) m ust b e ﬁnitely generated, or else it w ould b e uncountable. Next one uses that fo r a to rsion group T , Ext( T , Z ) is the P o n tr yagin dual of T, where T is giv en the discrete top ology , [15]. Th us, Ext( tK ∗ ( A ) , Z ) is a compact top ological group. If it is inﬁnite, then it is a p erfect top o logical space, hence uncountable, so it m ust b e ﬁnite. Therefore, K ∗ ( A ) is ﬁnitely generated. Similar ly , K ∗ ( A ) is ﬁnitely generated.  The Ruelle algebras R S ( X , ϕ, Q ) and R U ( X , ϕ, P ) are separable C ∗ -algebras asso ciated to amenable g roup oids. Hence, b y [44], they satisfy the Univ ersal Co eﬃcien t Theorem and (d) applies. Thus, w e o btain Prop osition 4.1. Th e gr oups K ∗ ( R S ( X , ϕ, Q )) , K ∗ ( R U ( X , ϕ, P )) , K ∗ ( R S ( X , ϕ, Q )) , and K ∗ ( R U ( X , ϕ, P )) ar e ﬁnitely gen er ate d . 14 JER OME KAMINKER , IAN F. PUTNAM 1 , AND MICHAEL F. WH ITT AKER Prop osition 4.2. If rank( K 0 ( R S ( X , ϕ, Q ))) = rank( K 1 ( R S ( X , ϕ, Q ))) , then R S ( X , ϕ, Q ) ∼ = R U ( X , ϕ, P ) . Pr o o f . The alg ebras R S ( X , ϕ, Q ) and R U ( X , ϕ, P ) satisfy the hypothesis of the Kirc h b erg- Phillips theorem, [28]. Th us we m ust only show that their K-theory gro ups are isomorphic. Considering torsion ﬁrst, note t ha t , using dualit y and the Unive rsal Co eﬃcien t Theorem, tK 0 ( R S ( X , ϕ, Q )) ∼ = Ext( K 1 ( R S ( X , ϕ, Q )) , Z ) ∼ = tK 1 ( R S ( X , ϕ, Q )) ∼ = tK 0 ( R U ( X , ϕ, P )) . F o r the free part, rank( K 0 ( R S ( X , ϕ, Q ))) = rank(Hom( K 0 ( R S ( X , ϕ, Q ))) , Z ) = rank(Hom( K 1 ( R U ( X , ϕ, P )) , Z )) = rank( K 1 ( R U ( X , ϕ, P ))) = rank( K 0 ( R U ( X , ϕ, P ))) A similar argumen t sho ws that K 1 ( R S ( X , ϕ, Q )) ∼ = K 1 ( R U ( X , ϕ, P )). Th us, by the classiﬁ- cation theorem, [2 8 ], w e obtain that R S ( X , ϕ, Q ) ∼ = R U ( X , ϕ, P ).  R e m ark. Note that S ( X , ϕ, Q ) and U ( X , ϕ, P ) ar e not isomorphic in general. Corollary 4.3. The algebr as R S ( X , ϕ, Q ) an d R U ( X , ϕ, P ) satisfy Poinc a r´ e duality. Pr o o f . The isomorphism b et w een R S ( X , ϕ, Q ) and R U ( X , ϕ, P ) is implem en t ed by an ele- men t ξ ∈ K K ( R S ( X , ϕ, Q ) , R U ( X , ϕ, P )). Consider the duality classes ∆ ∈ K K 1 ( R S ( X , ϕ, Q ) ⊗ R U ( X , ϕ, P ) , C ) and δ ∈ K K 1 ( C , R S ( X , ϕ, Q ) ⊗ R U ( X , ϕ, P )). Let ˜ δ ∈ K K 1 ( C , R U ( X , ϕ, P ) ⊗ R U ( X , ϕ, P )) and ˜ ∆ ∈ K K 1 ( R U ( X , ϕ, P ) ⊗ R U ( X , ϕ, P ) , C ) b e deﬁned b y (8) ˜ δ = δ τ ⊗ R S ( X,ϕ,Q ) ξ and (9) ˜ ∆ = ξ − 1 ⊗ R S ( X,ϕ,Q ) ∆ τ Then it follows from the prop erties of δ and ∆ tha t , for x ∈ K 0 ( R U ( X , ϕ, P )) one has (10) ˜ δ ⊗ R U ( X,ϕ,P ) ( x ⊗ R U ( X,ϕ,P ) ˜ ∆) = x.  The question of when the hy p othesis of Prop o sition 4.2 holds will no w b e addressed. Prop osition 4.4. Assume that K ∗ ( S ( X , ϕ , Q )) and K ∗ ( U ( X , ϕ, P )) hav e ﬁnite r ank. Then, one has rank K 0 ( R S ( X , ϕ, Q )) = rank K 1 ( R S ( X , ϕ, Q )) rank K 0 ( R U ( X , ϕ, P )) = rank K 1 ( R U ( X , ϕ, P )) . Pr o o f . W e will w ork out the case o f R U ( X , ϕ, P ), the case o f R S ( X , ϕ, Q ) b eing similar. K-THEORETIC DU ALITY FOR HYPERBOLIC DYNAMICAL SY STEMS 15 Consider the Pimsner-V oiculescu sequence tensored with Q , K 0 ( U ( X , ϕ, P )) ⊗ Q 1 − φ 0 ✲ K 0 ( U ( X , ϕ, P )) ⊗ Q α ✲ K 0 ( R U ( X , ϕ, P )) ⊗ Q K 1 ( R U ( X , ϕ, P )) ⊗ Q δ 1 ✻ ✛ β K 1 ( U ( X , ϕ, P )) ⊗ Q ✛ 1 − φ 1 K 1 ( U ( X , ϕ, P )) ⊗ Q . δ 0 ❄ One c hec ks directly that (11) K 0 ( R U ( X , ϕ, P )) ⊗ Q = image α ⊕ cok ernel α = cok ernel(1 − φ 0 ) ⊕ k er(1 − φ 1 ) Similarly , (12) K 1 ( R U ( X , ϕ, P )) ⊗ Q = cok ernel (1 − φ 1 ) ⊕ k er(1 − φ 0 ) . But one o bt a ins from, 0 → k er(1 − φ 0 ) → K 0 ( U ( X , ϕ, P )) ⊗ Q 1 − φ 0 − − − → K 0 ( U ( X , ϕ, P )) ⊗ Q → cok ernel(1 − φ 0 ) → 0 that (13) K 0 ( U ( X , ϕ, P )) ⊗ Q = imag e(1 − φ 0 ) ⊕ cok ernel (1 − φ 0 ) = image(1 − φ 0 ) ⊕ k er(1 − φ 0 ) . Since K 0 ( U ( X , ϕ, P )) is assumed to b e of ﬁnite ra nk it follows that rank(k er (1 − φ 0 )) = rank(cok ernel (1 − φ 0 )) a nd similarly rank(ker(1 − φ 1 )) = rank(cok ernel (1 − φ 1 )). Plugging these in to (1 1) and (12) yields the result.  4.5. Dynamical dualit y and the Baum-Connes conjecture. There are relations b e- t wee n the noncomm utativ e dualit y w e ha v e b een discussing and the Baum-Connes and No viko v conjectures in top olog y . W e consider a setting in whic h precise statemen ts can b e made. Let Γ b e a torsion free, ﬁnitely presen ted group. In this case, t he Baum-Connes map, after tensoring with Q , can b e iden tiﬁed with (14) µ ⊗ Q : K ∗ ( B Γ) ⊗ Q → K ∗ ( C ∗ r (Γ)) ⊗ Q , where the map µ ⊗ Q is obta ined by taking Kasparov pro duct with the class of the Mishc henk o line bundle δ Γ ∈ K K ( C , C 0 ( B Γ) ⊗ C ∗ r (Γ)). Th us, rational v ersion o f the Baum-Connes conjecture here is equiv alen t to C 0 ( B Γ) b eing (rationally) a Spanier- Whitehead dual to the noncomm utative algebra C ∗ r (Γ). Strictly sp eaking, it is the Baum- Connes conjecture as obtained by the Dirac-dual D irac metho d, since just hav ing t he ma p µ ⊗ Q an isomorphism migh t not imply the existence of the K-homolog y duality class. In many cases where the Baum-Connes conjecture has b een prov ed, use is made of non- p ositiv e curv a t ure. This often pro vides what is needed to deﬁne a K-homolo g y duality class whic h will give an in verse to the Baum-Connes map. The duality we study in the presen t pap er is based on h yp erb olic dynamics. On the other hand, in [8], and man y other w o rks, injectivit y is pro v ed for h yp erb olic g roups. Indeed, there is a relation b etw een the dualit y obtained here from h yp erb olic dynamics and that from h yp erb olic groups. Results in this 16 JER OME KAMINKER , IAN F. PUTNAM 1 , AND MICHAEL F. WH ITT AKER direction ha v e b een work ed out in the thesis of Emerson and the pap er b y Higson, [1 8]. W e will state a result in a sp ecial case. 4.4. Theorem ([13, 18, 42, 1]) . L et Γ b e a F uchsi a n gr oup with such that D / Γ is a c o mp act, oriente d surfac e, and supp ose that the b o unda ry of Γ is S 1 . Then one has the fol low ing c ommutative diagr am. K K Γ ( C 0 ( D ) , C ) µ ✲ K K ( C , C ∗ r (Γ)) K K 1 ( C ( ∂ Γ) ⋊ Γ , C ) ∂ ❄ E ⊗ C ( ∂ Γ) ⋊ Γ ✲ K K ( C , C ( ∂ Γ) ⋊ Γ) i ∗ ❄ wher e E ∈ K K 1 ( C ( ∂ Γ) ⋊ Γ) ⊗ C ( ∂ Γ) ⋊ Γ) , C ) is the elemen t c ons tructe d in [13] . Although the low er map w a s motiv ated b y the dynamical dualit y , the explicit connection is not apparen t. In the case at hand, a nd p ossibly in m uch more g eneralit y , it follows from [42] or [1], that there is a Smale space ( X , ϕ ) whose R uelle a lgebras are isomorphic to the crossed pro duct C ( ∂ Γ) ⋊ Γ. This can b e v eriﬁed by using the Kirch b erg-Phillips theorem and computing the K- t heory groups. What is not y et know n is whether there a re naturally deﬁned isomorphisms for the vertical ar r o ws making the diagram b elow comm utat ive. K K 1 ( C ( ∂ Γ) ⋊ Γ , C ) ✲ K K ( C , C ( ∂ Γ) ⋊ Γ) K K 1 ( R U ( X , ϕ, P ) , C )) ∼ = ❄ ✲ K K ( C , R S ( X , ϕ, Q )) , ∼ = ❄ Note that , once an isomorphism is c ho sen for the left arr ow, then there is a corresp onding one deﬁned fo r the right, but at presen t there is no geometrical w ay to obtain them. W e note that S. G orokho vsky , in his thesis, sho w ed that Pas c hke duality also ﬁts in t he presen t setting. Indeed, the Pasc hke dual of a C ∗ -algebra A is a Spanier-Whitehead dual in the sense describ ed here. 5. The K- theor y duality c lass W e giv e a desc ription of the dualit y class δ in K K ( S , R S ( X , ϕ, Q ) ⊗ R U ( X , ϕ, P )) for the Ruelle algebras. Let P and Q b e ϕ -in v ar ia n t sets of p erio dic p oin ts with P ∩ Q = ∅ . Before w e b egin with the tec hnical details, let us explain the underlying idea of the con- struction. Consider the pro duct g r o up oid G s ( X , ϕ, Q ) × G u ( X , ϕ, P ) whic h is equiv a lent to the group oid G h ( X , ϕ ) in the sense of Muhly , Renault, and Williams [2 5]. Since the gro up oid G h ( X , ϕ ) is an ´ etale group oid with compact unit space, namely X itself, its group oid C ∗ - algebra, H ( X , ϕ ), is unital. Th us, K 0 ( H ( X , ϕ )) has a cano nical elemen t determined by the class of the iden tit y . The ab ov e equiv alence o f g r o up oids implies that S ( X , ϕ, Q ) ⊗ U ( X , ϕ, P ) is Morita equiv a len t to H ( X , ϕ ) and we construct a pro j ection in S ( X , ϕ, Q ) ⊗ U ( X , ϕ, P ) K-THEORETIC DU ALITY FOR HYPERBOLIC DYNAMICAL SY STEMS 17 corresp onding to the class of the iden tit y in H ( X , ϕ ). F or details regar ding t he Morita equiv alence ab ov e see [3 1 ]. 5.1. Deﬁnition. Supp o se that F = { f 1 , f 2 , . . . , f K } are con tin uous, non-negativ e functions on X and G = { g 1 , · · · , g K } is a subset of X h ( P , Q ) = X s ( P ) ∩ X u ( Q ). F or 0 < ǫ ≤ ε ′ X , w e sa y that ( F , G ) is a n ε -partition of X if (1) the squares of the functions in F form a partit io n of unit y in C ( X ); that is, K X k =1 f 2 k = 1 , (2) the elemen ts of G are all distinct, (3) the supp ort of f k is con tained in B ( g k , ε/ 2), for eac h 1 ≤ k ≤ K . 5.2. Lemma. Ther e exists ( F , G ) , an ε ′ X -p artition of X such that ( F ◦ ϕ − 1 , ϕ ( G )) = ( { f k ◦ ϕ − 1 | 1 ≤ k ≤ K } , { ϕ ( g k ) | 1 ≤ k ≤ K } ) is also an ε ′ X -p artition of X . Mor e over, G c an b e chosen so that G ∩ ϕ ( G ) = ∅ . Pr o o f . Cho ose ε ′ X > ε ′ > 0 small enough that, fo r an y x in X , ϕ ( B ( x, ε ′ / 2)) ⊆ B ( ϕ ( x ) , ε ′ X / 2). Let U x = B ( x, ε ′ / 4) so tha t { U x } x ∈ X co vers X . Since X is compact there is a ﬁnite sub cov er, sa y { U k } K k =1 . Now a partition of unit y sub ordinate to { U k } K k =1 exists [3] and w e deﬁne F = { f 1 , f 2 , · · · , f K } to b e the square ro ots of these functions. Since X h ( P , Q ) is dense, we ma y c ho ose p oin ts g k in X h ( P , Q ) t o b e within ε ′ / 4 from the cen ter o f eac h ball U k . Now the supp ort of eac h function in F is still contained in a ball of radius ε ′ / 2. Therefore, w e ha ve an ε ′ X -partition ( F , G ) suc h tha t ( F ◦ ϕ − 1 , ϕ ( G )) is also an ε ′ X -partition.  No w, for 0 < ε ≤ ε ′ X , let ( F , G ) b e an ε -partition and deﬁne a function p G on G s ( X , ϕ, Q ) × G u ( X , ϕ, P ) b y setting p G (( x, x ′ ) , ( y , y ′ )) = f i ([ x, y ]) f j ([ x ′ , y ′ ]) , for ( x, x ′ ) ∈ G s ( X , ϕ, Q ) , ( y , y ′ ) ∈ G u ( X , ϕ, P ), if, for some i, j , x ∈ X u ( g i , ε ) , y ∈ X s ( g i , ε ) , x ′ ∈ X u ( g j , ε ) , y ′ ∈ X s ( g j , ε ) , [ x, y ] = [ x ′ , y ′ ] and to b e zero otherwise. Notice that if a pair i, j exist f o r a giv en (( x, x ′ ) , ( y , y ′ )), then it is unique, since g i = [ y , x ] and g j = [ y ′ , x ′ ]. 5.3. Lemma. L et 0 < ε ≤ ε ′ X and let ( F , G ) b e an ǫ -p artition. Then p G is i n S ( X , ϕ, Q ) ⊗ U ( X , ϕ, P ) . Pr o o f . Let us ﬁx a pair i, j and supp ose there exists ( x, x ′ ) ∈ G s ( X , ϕ, Q ) and ( y , y ′ ) ∈ G u ( X , ϕ, P ) suc h that x ∈ X u ( g i , ε ) , y ∈ X s ( g i , ε ) , x ′ ∈ X u ( g j , ε ) , y ′ ∈ X s ( g j , ε ) , [ x, y ] = [ x ′ , y ′ ] . W e note that [ g i , g j ] is deﬁned and is stably equiv a len t to g i and unstably equiv alen t to g j . By lemma 3.1 there are lo cal ho meomorphisms h s : X u ( g i , ε ) → X u ([ g i , g j ] , ε ) and h u : X s ( g j , ε ) → X s ([ g i , g j ] , ε ) deﬁned by h s ( x ) = [ x, [ g i , g j ]] = [ x, g j ] h u ( y ′ ) = [[ g i , g j ] , y ′ ] = [ g i , y ′ ] 18 JER OME KAMINKER , IAN F. PUTNAM 1 , AND MICHAEL F. WH ITT AKER It is immediate that if w e let x ′ = h s ( x ) a nd y = h u ( y ′ ) then the p o in ts satisfy the conditions ab ov e. On t he other hand, if (( x, x ′ ) , ( y , y ′ )) satisfy the conditions then w e ha ve x ′ = [[ x ′ , y ′ ] , x ′ ] = [[ x, y ] , x ′ ] = [ x, x ′ ] = [ x, g j ] = h s ( x ) y = [ y , [ x, y ]] = [ y , [ x ′ , y ′ ]] = [ y , y ′ ] = [ g i , y ′ ] = h u ( y ′ ) . This sho ws that p oints satisfying the conditions are realized b y lo cal homeomorphisms, one on the lo cal unstable set of g i and one o n the lo cal stable set of g j . Set ε ′ > 0. Consider the function on X u ( g i , ε ) × X s ( g j , ε ) sending ( x, y ′ ) to f i ([ x, y ]) f j ([ x ′ , y ′ ]). It is clearly a con tin uous function of compact supp o rt so that it can b e uniformly approx i- mated within ε ′ b y a function of the form K i,j X k =1 a i,j,k ( x, x ′ ) b i,j,k ( y , y ′ ) where, for each ﬁxed k , w e ha v e a i,j,k in C c ( G s ( X , ϕ, Q )) and b i,j,k in C c ( G u ( X , ϕ, P )). If there exists no (( x, x ′ ) , ( y , y ′ )) f o r a ﬁxed i, j we deﬁne the ab o ve sum to b e zero. No w it follo ws that X i,j K i,j X k =1 a i,j,k ⊗ b i,j,k is within ε ′ of p G in norm. This completes the pro of.  In the sequel, it will b e con venie n t to hav e a description o f the o p erator p G on the Hilb ert space ℓ 2 ( X h ( P , Q )) ⊗ ℓ 2 ( X h ( P , Q )), in terms of our usual basis, { δ w ⊗ δ z | w , z ∈ X h ( P , Q ) } . W e also intro duce a standard con ven t ion that the bra ck et map returns the empt y set when the brack et of tw o p oin ts is undeﬁned. Of course, any op erato r applied to the dirac delta function of the empt y set will return zero and w e declare that a ny function of the empt y set is also zero. This con v ention will simplify many of the up coming form ulations. 5.4. Lemma. L et 0 < ε ≤ ε ′ X and let ( F , G ) b e an ε -p a rtition. Supp o s e w , z ar e in X h ( P , Q ) , then we have p G ( δ w ⊗ δ z ) = f k ([ w , z ]) K X i =1 f i ([ w , z ]) δ [ w ,g i ] ⊗ δ [ g i ,z ] if ther e exi s ts a 1 ≤ k ≤ K , such that w ∈ X u ( g k , ε ) , z ∈ X s ( g k , ε ) and i s zer o if ther e is no such k . (If the k exists, it is unique, for given w , z . The expr essio n on the right makes sense using our standa r d c on vention). Pr o o f . F or an y x, y in X h ( P , Q ), we compute ( p G ( δ w ⊗ δ z ))( x, y ) = X x ′ ∈ X h ( x ) X y ′ ∈ X h ( y ) p G (( x, x ′ ) , ( y , y ′ )) δ w ( x ′ ) δ z ( y ′ ) = p G (( x, w ) , ( y , z )) = f i ([ x, y ]) f k ([ w , z ]) , pro vided x ∈ X u ( g i , ε ) , y ∈ X s ( g i , ε ) , w ∈ X u ( g k , ε ) , z ∈ X s ( g k , ε ) , [ x, y ] = [ w , z ] K-THEORETIC DU ALITY FOR HYPERBOLIC DYNAMICAL SY STEMS 19 and zero otherwise. If t here is no k suc h that w ∈ X u ( g k , ε ) , z ∈ X s ( g k , ε ), then the conclusion holds. Let us con tin ue under the assumption that t here is suc h a k (whic h m ust b e unique , since [ z , w ] = g k and the brack et map is (lo cally) unique in a Smale space ). If, for some i , [ w , z ] is not in the supp ort of f i , then fo r an y x, y as ab ov e for whic h [ x, y ] = [ w , z ], w e ha v e f i ([ x, y ]) = f i ([ w , z ]) = 0. On the other hand, if [ w , z ] is in the supp or t of f i , fo r some i , then x = [ x, g i ] = [[ x, y ] , g i ] = [[ w , z ] , g i ] = [ w , g i ] y = [ g i , y ] = [ g i , [ x, y ]] = [ g i , [ w , z ]] = [ g i , z ] . That is, for a giv en i , the c hoice of x, y is unique. F or each suc h i , w e hav e ( p G ( δ w ⊗ δ z ))([ w , g i ] , [ g i , z ]) = f i ([ w , z ]) f k ([ w , z ]) , and the left hand side is zero for all other v a lues of x, y . The conclusion follows.  5.5. Lemma. L e t 0 < ε ≤ ε ′ X . I f ( F , G ) is an ε -p artition, then p G is a pr oje ction. If ( F ◦ ϕ − 1 , ϕ ( G )) is al s o a n ε -p artition, then ( u ⊗ u )( p G )( u ∗ ⊗ u ∗ ) = p ϕ ( G ) . Pr o o f . T o sho w that p G is a pro j ection w e use lemma 5 .4 to compute ( p G ) 2 ( δ w ⊗ δ z ). F irst of all, we hav e p G ( δ w ⊗ δ z ) = f k ([ w , z ]) K X i =1 f i ([ w , z ]) δ [ w ,g i ] ⊗ δ [ g i ,z ] , if w ∈ X u ( g k , ε ) , z ∈ X s ( g k , ε ) and zero otherwise. W e apply p G again, taking it throug h the sum and lo oking at each term individually . That is, for ﬁxed 1 ≤ i ≤ K , w e m ust consider, for what l is [ w , g i ] in X u ( g l , ε ) and [ g i , z ] in X s ( g l , ε ). Since [ w , g i ] is clearly in X u ( g i ), this can only happ en for l = i . Using this, w e obtain ( p G ) 2 ( δ w ⊗ δ z ) = f k ([ w , z ]) K X i =1 f i ([ w , z ]) p G δ [ w ,g i ] ⊗ δ [ g i ,z ] = f k ([ w , z ]) K X i =1 f i ([ w , z ]) f i ([ w , z ]) K X j =1 f j ([[ w , g i ] , [ g i , z ]]) δ [[ w ,g i ] ,g j ] ⊗ δ [ g j , [ g i ,z ]] = f k ([ w , z ]) K X i =1 f i ([ w , z ]) 2 K X j =1 f j ([ w , z ]) δ [ w ,g j ] ⊗ δ [ g j ,z ] = f k ([ w , z ]) K X j =1 f j ([ w , z ]) δ [ w ,g j ] ⊗ δ [ g j ,z ] = p G ( δ w ⊗ δ z ) The second part of the pro of is a computation and is omitted.  F r o m Lemma 5.2, we may ﬁnd F = { f 1 , . . . , f K } , G = { g 1 , . . . , g K } suc h that ( F , G ) and ( F ◦ ϕ − 1 , ϕ ( G )) are b oth ε ′ X -partitions of X with G ∩ ϕ ( G ) = ∅ . Since X h ( P , Q ) contains no p erio dic p oin t s we kno w that neither do es G . By lemmas 5.3 and 5.5, w e hav e that b oth 20 JER OME KAMINKER , IAN F. PUTNAM 1 , AND MICHAEL F. WH ITT AKER p G and p ϕ ( G ) are pro jections in S ( X , ϕ, Q ) ⊗ U ( X , ϕ, P ). F or each 0 ≤ s ≤ 1, consider the collection F s = { (1 − s ) 1 / 2 f 1 , . . . , (1 − s ) 1 / 2 f K , s 1 / 2 f 1 ◦ ϕ − 1 , . . . , s 1 / 2 f K ◦ ϕ − 1 } together with the set of p o ints G s = { g 1 , . . . , g K , ϕ ( g 1 ) , . . . , ϕ ( g K ) } Clearly , ( F s , G s ) is an ε ′ X -partition, for all 0 ≤ s ≤ 1. The imp ortant features of p G s are (1) p G s is a pat h o f pro j ections in S ( X , ϕ, Q ) ⊗ U ( X , ϕ, P ), (2) p G s arises from the ε ′ X -partition ( F s , G s ), for all 0 ≤ s ≤ 1, (3) p G 0 = p G and (4) p G 1 = ( u ⊗ u ) p G ( u ∗ ⊗ u ∗ ) = p ϕ ( G ) . Therefore p G and p ϕ ( G ) are homotopic pro jections in S ( X , ϕ, Q ) ⊗ U ( X , ϕ, P ). Since p G and p ϕ ( G ) are homotopic, there is a partial isometry v in S ( X , ϕ, Q ) ⊗ U ( X , ϕ, P ) with initial pro jection v ∗ v = p G and ﬁnal pro jection v v ∗ = p ϕ ( G ) . By lemma 5.5 w e ha ve that ( u ⊗ u ) p G ( u ∗ ⊗ u ∗ ) = p ϕ ( G ) and it is easy to chec k t ha t the op erator  = ( u ⊗ u ) p G v ∗ has the prop ert y  ∗  =  ∗ = p ϕ ( G ) . Note that the op erato r  is in R S ( X , ϕ, Q ) ⊗ R U ( X , ϕ, P ) but not in S ( X , ϕ, Q ) ⊗ U ( X , ϕ, P ) since u ⊗ u is in the former but not the la tter. W e are now ready to deﬁne a ∗ -homomorphism δ : S → R S ( X , ϕ, Q ) ⊗ R U ( X , ϕ, P ). T o do this, it suﬃce s to deﬁne a partial isometry V in R S ( X , ϕ, Q ) ⊗ R U ( X , ϕ, P ) with the same initial a nd ﬁnal pro jection, V ∗ V = V V ∗ is a pro jection. Then sending z − 1 to V − V ∗ V extends uniquely to suc h a map. (T o see this, w e simply note that V + ( 1 − V ∗ V ) is a unitary in the unitization of the range. So there is a unique ∗ -homomorphism mapping z in C ( S 1 ) to V , whose restriction to S ∼ = C ∗ ( z − 1) is as claimed). Since  in R S ( X , ϕ, Q ) ⊗ R U ( X , ϕ, P ) has the prop ert y that  ∗  =  ∗ = p ϕ ( G ) , w e obta in the required ∗ -homomor phism, whic h w e denote by δ . 5.6. Deﬁnition. The class δ in K K ( S , R S ( X , ϕ, Q ) ⊗ R U ( X , ϕ, P )) is deﬁned b y the ∗ - homomorphism δ fro m S to R S ( X , ϕ, Q ) ⊗ R U ( X , ϕ, P ) whic h is uniquely determined b y δ ( z − 1) =  −  ∗  where  = ( u ⊗ u ) p G v ∗ . 6. The K-homology duality class F o r the K-homology dualit y class we construct a n extension of R S ( X , ϕ, Q ) ⊗ R U ( X , ϕ, P ). Recall that H = ℓ 2 ( X h ( P , Q )). F rom section 3, w e hav e represen ta tions of S ( X , ϕ, Q ), U ( X , ϕ, P ), R S ( X , ϕ, Q ) and R U ( X , ϕ, P ) as b ounded op erato rs on H . The ﬁrst observ ation is that, since these alg ebras are r epresen ted on the same Hilb ert space, we can consider how op erators coming fr o m S ( X , ϕ , Q ) a nd U ( X , ϕ, P ) interact on H . The follo wing three L emmas elucidate these in teractions. W e hav e used a h yp erb olic toral automo r phism to illustrated the main concepts in Lemma 6.1 and Lemma 6.3 in Fig ur es 3 and 4, resp ectiv ely . F o r these three Lemmas let us ﬁx the following elemen ts. Assume that a in S ( X , ϕ, Q ) and b in U ( X , ϕ, P ) are b oth supp orted on basic sets; that is, for v , w ∈ X u ( Q ) and v ′ , w ′ ∈ X s ( P ), let the supp ort of a b e V s ( v , w, h s , δ ) and the supp ort of b b e V u ( v ′ , w ′ , h u , δ ′ ). Note K-THEORETIC DU ALITY FOR HYPERBOLIC DYNAMICAL SY STEMS 21 that S ou r ce ( a ) ⊆ X u ( w , δ ) and Rang e ( a ) ⊆ X u ( v , δ ) , and S our ce ( b ) ⊆ X s ( w ′ , δ ′ ) and Rang e ( b ) ⊆ X s ( v ′ , δ ′ ). See lemma 3.3 for further details. 6.1. Lemma ( [31]) . If a is in S ( X , ϕ, Q ) and b is in U ( X , ϕ, P ) , then ab and ba ar e c omp act op er ators on H . Pr o o f . W e compute, fo r x in X h ( P , Q ), a · b δ x = a ( h s ◦ h u ( x ) , h u ( x )) b ( h u ( x ) , x ) δ h s ◦ h u ( x ) if x ∈ X s ( w ′ , δ ′ ), h u ( x ) ∈ X s ( v ′ , δ ′ ), h u ( x ) ∈ X u ( w , δ ), and h s ◦ h u ( x ) ∈ X u ( v , δ ) . Ot herwise the pro duct is zero. In particular, the pro duct is zero unless R ang e ( b ) ∩ S our ce ( a ) is non- zero. Ho w eve r, uniqueness of the bra c ket implies tha t a lo cal stable set and a lo cal unstable set hav e non-trivial in t ersection at one p oint, at most. Whence, the pro duct is zero unless X s ( v ′ , δ ′ ) and X u ( w , δ ) inters ect and if they do the pro duct is a rank o ne op erat o r. Now ﬁnite sums o f operat ors with supp orts a s ab ov e f o rm a dense set and therefore we obtain the compact op erato rs b y taking limits. T aking adjoints giv es that b · a is also compact.  (0 , 0) (0 , 1) (1 , 0) (1 , 1) X u ( w , δ ) X u ( v , δ ) h s w v h u ( x ) h s ◦ h u ( x ) X s ( v ′ , δ ′ ) X s ( w ′ , δ ′ ) h u v ′ w ′ x Figure 3. Hy p erb olic tora l a utomorphism: ab is a compact o p erator 6.2. Lemma. If a is in S ( X , ϕ, Q ) an d b is in U ( X , ϕ, P ) , then lim n → + ∞ α − n s ( a ) · b = 0 and lim n → + ∞ b · α − n s ( a ) = 0 . Pr o o f . W e ﬁrst aim to show that there exists N in N suc h that, for all n ≥ N , we ha v e α − n ( a ) · b = 0. Indeed, since α − n s ( a ) δ z = a ( h s ◦ ϕ n ( z ) , ϕ n ( z )) δ ϕ − n ◦ h s ◦ ϕ n ( z ) , w e see that the supp or t of α − n s ( a ) is V s ( ϕ − n ( v ) , ϕ − n ( w ) , ϕ − n ◦ h s ◦ ϕ − n , λ − n δ ). Therefore, it follo ws that S our ce ( α − n s ( a )) ⊆ X u ( ϕ − n ( w ) , λ − n δ ) and Rang e ( α − n s ( a )) ⊆ X u ( ϕ − n ( v ) , λ − n δ ). 22 JER OME KAMINKER , IAN F. PUTNAM 1 , AND MICHAEL F. WH ITT AKER That is, t he supp ort of a is being exp o nen tially con tra cted b y rep eated applicatio n of α s . Moreo ver, w e compute α − n s ( a ) · b δ z = a ( h s ◦ ϕ n ◦ h u ( z ) , ϕ n ◦ h u ( z )) b ( h u ( z ) , z ) δ ϕ − n ◦ h s ◦ ϕ n ◦ h u ( z ) if z ∈ X s ( w ′ , δ ′ ), h u ( z ) ∈ X s ( v ′ , δ ′ ), and h u ( z ) ∈ X u ( ϕ − n ( w ) , λ − n δ ). It is zero otherwise. No w set ε > 0 small enough that X u ( Q, ε ) ∩ X s ( v ′ , δ ′ ) = ∅ , we kno w this is p ossible since v ′ is in X s ( P ) while no p oin t in Q is in X s ( P ) since P and Q are m utually distinct and ϕ -in v a r ia n t. Giv en ε > 0, we can ﬁnd a n N in N , suc h that X u ( ϕ − n ( w ) , λ − n δ ) ⊂ X u ( Q, ε ) for all n ≥ N . This implies that, for a ll n ≥ N , w e hav e α − n ( a ) · b = 0. No w the general r esult follo ws since elemen ts of S ( X , ϕ, Q ) and U ( X , ϕ, P ) ar e norm limits of linear combin ations of elemen ts with the ab ov e f o rm. A similar argumen t giv es the result for b · α − n ( a ).  6.3. Lemma ([3 1] [22]) . F or any a in S ( X , ϕ , Q ) and b in U ( X , ϕ, P ) , we have lim n →∞ k α n s ( a ) b − bα n s ( a ) k = 0 , lim n →∞ k α n s ( a ) α − n u ( b ) − α − n u ( b ) α n s ( a ) k = 0 . Pr o o f . W e shall prov e the second equality only , from whic h the ﬁrst is easily deduced. Set ε > 0. W e compute α n s ( a ) · α − n u ( b ) δ z = a ( h s ◦ ϕ − 2 n ◦ h u ◦ ϕ n ( z ) , ϕ − 2 n ◦ h u ◦ ϕ n ( z )) b ( h u ◦ ϕ n ( z ) , ϕ n ( z )) δ ϕ n ◦ h s ◦ ϕ − 2 n ◦ h u ◦ ϕ n ( z ) α − n u ( b ) · α n s ( a ) δ z = b ( h u ◦ ϕ 2 n ◦ h s ◦ ϕ − n ( z ) , ϕ 2 n ◦ h s ◦ ϕ − n ( z )) a ( h s ◦ ϕ − n ( z ) , ϕ − n ( z )) δ ϕ − n ◦ h u ◦ ϕ 2 n ◦ h s ◦ ϕ − n ( z ) . Moreo ver, lemma 2 . 2 in [31] states tha t , there exists N suc h that for all n ≥ N we ha ve, ϕ n ◦ h s ◦ ϕ − 2 n ◦ h u ◦ ϕ n ( z ) = ϕ − n ◦ h u ◦ ϕ 2 n ◦ h s ◦ ϕ − n ( z ) . No w, supp o se w e are given z in X h ( P , Q ) suc h that ϕ − n ( z ) ∈ S our ce ( a ) and ϕ n ( z ) ∈ S our ce ( b ). W e ma y deﬁne the follo wing p o in ts: x 1 = z x 2 = ϕ n ◦ h s ◦ ϕ − n ( z ) x 3 = ϕ n ◦ h s ◦ ϕ − 2 n ◦ h u ◦ ϕ n ( z ) = ϕ − n ◦ h u ◦ ϕ 2 n ◦ h s ◦ ϕ − n ( z ) x 4 = ϕ − n ◦ h u ◦ ϕ n ( z ) . In fact, giv en ε 1 > 0 and any z satisfying the ab o v e conditions, we can set N suﬃcien tly large that we hav e, fo r all n ≥ N : x 2 ∈ X s ( x 1 , ε 1 ) x 4 ∈ X u ( x 1 , ε 1 ) , x 1 ∈ X s ( x 2 , ε 1 ) x 3 ∈ X u ( x 2 , ε 1 ) , x 4 ∈ X s ( x 3 , ε 1 ) x 2 ∈ X u ( x 3 , ε 1 ) , x 3 ∈ X s ( x 4 , ε 1 ) x 1 ∈ X u ( x 4 , ε 1 ) . K-THEORETIC DU ALITY FOR HYPERBOLIC DYNAMICAL SY STEMS 23 W e ha ve illustrated the relationship b etw een these p oints for the hyperb olic toral automor- phism in F igure 4 on page 24. Since a and b are taking basis v ectors to ba sis v ectors, w e ha ve k α n s ( a ) · α − n u ( b ) − α − n u ( b ) · α n s ( a ) k = sup z | a ( ϕ − n ( x 3 ) , ϕ − n ( x 4 )) b ( ϕ n ( x 4 ) , ϕ n ( x 1 )) − b ( ϕ n ( x 3 ) , ϕ n ( x 2 )) a ( ϕ − n ( x 2 ) , ϕ − n ( x 1 )) | . No w a and b are unifo rmly con tinu ous so w e ma y c ho ose N la r g e enough that the a b ov e condition is satisﬁed (the t w o v ersions o f x 3 are equal) and so that ε 1 is suﬃcien tly small that w e hav e, for all n ≥ N , | a ( ϕ − n ( x 3 ) , ϕ − n ( x 4 )) − a ( ϕ − n ( x 2 ) , ϕ − n ( x 1 )) | < ε 2 k b k | b ( ϕ n ( x 4 ) , ϕ n ( x 1 )) − b ( ϕ n ( x 3 ) , ϕ n ( x 2 )) | < ε 2 k a k . No w w e compute | a ( ϕ − n ( x 3 ) , ϕ − n ( x 4 )) b ( ϕ n ( x 4 ) , ϕ n ( x 1 )) − a ( ϕ − n ( x 2 ) , ϕ − n ( x 1 )) b ( ϕ n ( x 3 ) , ϕ n ( x 2 )) | = | a ( ϕ − n ( x 3 ) , ϕ − n ( x 4 )) b ( ϕ n ( x 4 ) , ϕ n ( x 1 )) − a ( ϕ − n ( x 3 ) , ϕ − n ( x 4 )) b ( ϕ n ( x 3 ) , ϕ n ( x 2 )) + a ( ϕ − n ( x 3 ) , ϕ − n ( x 4 )) b ( ϕ n ( x 3 ) , ϕ n ( x 2 )) − a ( ϕ − n ( x 2 ) , ϕ − n ( x 1 )) b ( ϕ n ( x 3 ) , ϕ n ( x 2 )) | ≤ | a ( ϕ − n ( x 3 ) , ϕ − n ( x 4 )) || b ( ϕ n ( x 4 ) , ϕ n ( x 1 )) − b ( ϕ n ( x 3 ) , ϕ n ( x 2 )) | + | a ( ϕ − n ( x 3 ) , ϕ − n ( x 4 )) − a ( ϕ − n ( x 2 ) , ϕ − n ( x 1 )) || b ( ϕ n ( x 3 ) , ϕ n ( x 2 )) | < k a k ε 2 k a k + k b k ε 2 k b k = ε. Therefore, w e hav e show n that lim n →∞ k α n s ( a ) α − n u ( b ) − α − n u ( b ) α n s ( a ) k = 0 whic h completes the pro of.  This completes the interactions of S ( X , ϕ, Q ) and U ( X , ϕ, P ) on H . Our goa l is to pro duce an extension of R S ( X , ϕ, Q ) ⊗ R U ( X , ϕ, P ). T o accomplish this we shall represen t eac h of these C ∗ -algebras as op erators on a Hilb ert space such that they commute mo dulo compact op erators. Consider the Hilb ert space H = H ⊗ ℓ 2 ( Z ) = M n ∈ Z H . W e shall deﬁne represen tations o f R S ( X , ϕ, Q ) and R U ( X , ϕ, P ) as b ounded op erato rs on H and sho w that the in teraction of these algebras naturally giv es rise to an extension of R S ( X , ϕ, Q ) ⊗ R U ( X , ϕ, P ) by the compact op erators of H . Recall that fo r δ x in H w e ha v e the unitary op erator uδ x = δ ϕ ( x ) and α s ( a ) = uau ∗ and α u ( b ) = ubu ∗ . The bilateral shift o n ℓ 2 ( Z ) will b e denoted b y B and is the op erator giv en b y B δ n = δ n − 1 . W e note that from this p o int forw ar ds w e will alw a ys use δ m and δ n as basis 24 JER OME KAMINKER , IAN F. PUTNAM 1 , AND MICHAEL F. WH ITT AKER (0 , 0) (0 , 1) (1 , 0) (1 , 1) S ou r ce ( a ) Rang e ( a ) h u S ou r ce ( b ) Rang e ( b ) h s ϕ − n ( x 1 ) ϕ − n ( x 4 ) ϕ − n ( x 2 ) ϕ − n ( x 3 ) ϕ n ( x 1 ) ϕ n ( x 2 ) ϕ n ( x 4 ) ϕ n ( x 3 ) (0 , 0) (0 , 1) (1 , 0) (1 , 1) S ou r ce ( α n s ( a )) Rang e ( α n s ( a )) S ou r ce ( α − n u ( b )) Rang e ( α − n u ( b )) x 1 x 4 x 3 x 2 Figure 4. The p oints x 1 , x 2 , x 3 , x 4 for a h yp erb olic toral a uto morphism. v ectors of ℓ 2 ( Z ) a nd δ x , δ y and δ z as basis v ectors of H = ℓ 2 ( X h ( P , Q )). Finally , let us also deﬁne the op erator U = M n ∈ Z u n =        . . . u − 1 u 0 u 1 . . .        ∈ B ( H ⊗ ℓ 2 ( Z )) . (15) Deﬁne π s : R S ( X , ϕ, Q ) → B ( H ⊗ ℓ 2 ( Z )), for a in S ( X , ϕ, Q ), via π s ( a ) = M n ∈ Z α n s ( a ) = U ( a ⊗ 1) U ∗ π s ( u ) = 1 ⊗ B . Also deﬁne π u : R U ( X , ϕ, P ) → B ( H ⊗ ℓ 2 ( Z )), for b in U ( X , ϕ, P ), via π u ( b ) = b ⊗ 1 π u ( u ) = u ⊗ B ∗ . The reader is in vited to c hec k that t hese are co v ariant r epresen tations of the Ruelle alg ebras. W e now consider the interactions of S ( X , ϕ, Q ), U ( X, ϕ, P ), R S ( X , ϕ, Q ), and R U ( X , ϕ, P ) as op erato rs on H . 6.4. Lemma. F or any f in R S ( X , ϕ, Q ) an d g in R U ( X , ϕ, P ) , we have [ π s ( f ) , π u ( g )] = π s ( f ) π u ( g ) − π u ( g ) π s ( f ) is a c omp act op er ator on H . K-THEORETIC DU ALITY FOR HYPERBOLIC DYNAMICAL SY STEMS 25 Pr o o f . F rom lemma 6.1 w e know that on eac h coo rdinate of H , for a in S ( X , ϕ, Q ) and b in U ( X , ϕ, P ), w e ha v e π s ( a ) π u ( b ) and π u ( b ) π s ( a ) are compact op erators. Denote the n th co ordinate o f H = M n ∈ Z H b y H n and set ε > 0. Lemma 6.2 implies that there exists N 1 suc h tha t for n ≥ N 1 w e ha ve that b oth k α − n ( a ) b k < ε/ 2 and k bα − n ( a ) k < ε/ 2. Therefore, k ( π s ( a ) π u ( b ) − π u ( b ) π s ( a )) | H − n k = k α − n ( a ) b − bα − n ( a ) k ≤ k α − n ( a ) b k + k bα − n ( a ) k < ε. Moreo ver, Lemma 6.3 implies that there exists N 2 suc h that for n ≥ N 2 w e hav e k π s ( a ) π u ( b ) − π u ( b ) π s ( a )) | H n k = k α n ( a ) b − bα n ( a ) k < ε. Therefore, f o r a ∈ S ( X , ϕ, Q ) and b ∈ U ( X , ϕ, P ) w e ha v e [ π s ( a ) , π u ( b )] is compact. More- o v er, computations sho w that [ π s ( a ) , π u ( u )] = 0, [ π u ( b ) , π s ( u )] = 0, and [ π u ( u ) , π s ( u )] = 0. The conclusion follo ws.  The pro of of the next lemma is omitted other than to note that the result follows imme- diately from t he irreducibilit y o f the Smale space itself. 6.5. Lemma. If a is in S ( X , ϕ, Q ) and b is in U ( X , ϕ, P ) , then π s ( a ) π u ( b ) and π u ( b ) π s ( a ) ar e never c omp a c t op er ators on H unless ei ther a or b is the ze r o op er a tor. Deﬁne E to b e t he C ∗ -algebra generated b y π s ( R S ( X , ϕ, Q )), π u ( R U ( X , ϕ, P )), and K ( H ). Note that neither π s ( R S ( X , ϕ, Q )) or π u ( R U ( X , ϕ, P )) contain an y compact op erat ors on H other than t he zero op erator. Lemma 6.4 implies tha t π s ( R S ( X , ϕ, Q )) and π u ( R U ( X , ϕ, P )) comm ute mo dulo the compact o p erators K ( H ). F rom this w e hav e that E / K ( H ) ∼ = R S ( X , ϕ, Q ) ⊗ R U ( X , ϕ, P ) . Therefore, w e obtain an extension ∆ in K K 1 ( R S ( X , ϕ, Q ) ⊗ R U ( X , ϕ, P ) , C ). 6.6. Deﬁnition. The class ∆ in K K 1 ( R S ( X , ϕ, Q ) ⊗ R U ( X , ϕ, P ) , C ) is r epresen ted by the extension 0 ✲ K ( H ) ✲ E ✲ R S ( X , ϕ, Q ) ⊗ R U ( X , ϕ, P ) ✲ 0 . 7. Proo f of the main res ul t W e giv e a pro of of the main r esult, Theorem 1.1. W e fo cus our at t ention on sho wing that δ ⊗ R U ( X,ϕ,P ) ∆ = 1 R S ( X,ϕ,Q ) and note tha t an analogous argumen t sho ws that δ ⊗ R S ( X,ϕ,Q ) ∆ = − 1 R U ( X,ϕ,P ) . The pro of is divided in to roughly 3 parts. In the ﬁr st we describ e the elemen t δ ⊗ R U ( X,ϕ,P ) ∆ as an extension. In the second part, w e apply a ty p e of un t wisting to this extension. Finally , w e sho w that, up to unitary equiv alence and Bo tt p erio dicit y , the class w e hav e obtained is represen ted by 1 R S ( X,ϕ,Q ) . 7.1. Lemma. The cl a ss o f δ ⊗ R U ( X,ϕ,P ) ∆ in K K 1 ( R S ( X , ϕ, Q ) ⊗ S , R S ( X , ϕ, Q )) is given by the extension 0 ✲ R S ( X , ϕ, Q ) ⊗ K ( H ) ✲ E ′ σ ∗ ◦ π ′ ✲ R S ( X , ϕ, Q ) ⊗ S ✲ 0 . 26 JER OME KAMINKER , IAN F. PUTNAM 1 , AND MICHAEL F. WH ITT AKER Pr o o f . Recall that the expanded pro duct is δ ⊗ R U ( X,ϕ,P ) ∆ = σ ∗ ( τ S τ R S ( X,ϕ,Q ) ( δ ) ⊗ τ R S ( X,ϕ,Q ) σ ∗ (∆)) . The Kasparov pro duct is obtained b y comp osing the ∗ - ho momorphism τ S τ R S ( X,ϕ,Q ) ( δ ), whic h is giv en by δ ⊗ 1 : S ⊗ R S ( X , ϕ, Q ) − → R S ( X , ϕ, Q ) ⊗ R U ( X , ϕ, P ) ⊗ R S ( X , ϕ, Q ) , and the represen tation 1 ⊗ π u ⊗ π s : R S ( X , ϕ, Q ) ⊗ R U ( X , ϕ, P ) ⊗ R S ( X , ϕ, Q ) → B ( H ⊗ H ⊗ ℓ 2 ( Z )) deﬁning the extension τ R S ( X,ϕ,Q ) σ ∗ (∆), give n b y 0 → R S ( X , ϕ, Q ) ⊗K ( H ) → R S ( X , ϕ, Q ) ⊗E → R S ( X , ϕ, Q ) ⊗ R U ( X , ϕ, P ) ⊗ R S ( X , ϕ, Q ) → 0 . Whence, the class δ ⊗ R U ( X,ϕ,P ) ∆ is give n by the extension 0 ✲ R S ( X , ϕ, Q ) ⊗ K ( H ) ✲ E ′ ✲ R S ( X , ϕ, Q ) ⊗ S ✲ 0 where E ′ is the C ∗ -algebra generated by R S ( X , ϕ, Q ) ⊗ K ( H ) a nd the image of the map (1 ⊗ π u ⊗ π s ) ◦ ( δ ⊗ 1) ◦ σ ∗ : R S ( X , ϕ, Q ) ⊗ S → B ( H ⊗ H ⊗ ℓ 2 ( Z )) .  W e note that the ima g e of a · u k ⊗ z − 1 completely determines the map (1 ⊗ π u ⊗ π s ) ◦ ( δ ⊗ 1) ◦ σ ∗ : R S ( X , ϕ, Q ) ⊗ S → B ( H ⊗ H ⊗ ℓ 2 ( Z )) . Recall that b oth p G and v are elemen ts of R S ( X , ϕ, Q ) ⊗ R U ( X , ϕ, P ) (see L emma 5 .3 o n page 17). Moreo ver, let us extend the elemen t U , deﬁne d b y (15) on page 24, to H ⊗ H ⊗ ℓ 2 ( Z ) via U = M n ∈ Z u n ⊗ u n =        . . . u − 1 ⊗ u − 1 u 0 ⊗ u 0 u 1 ⊗ u 1 . . .        ∈ B ( H ⊗ H ⊗ ℓ 2 ( Z )) . No w we des crib e the map ( 1 ⊗ π u ⊗ π s ) ◦ ( δ ⊗ 1) ◦ σ ∗ : R S ( X , ϕ, Q ) ⊗ S → B ( H ⊗ H ⊗ ℓ 2 ( Z )) on generators: 1 ⊗ z 7→ (( u ⊗ u ) p G v ∗ ) ⊗ 1 , a ⊗ 1 7→ [(( u ⊗ u ) p G ( u ⊗ u ) ∗ ) ⊗ 1)][ U (1 ⊗ a ⊗ 1) U ∗ ] , u ⊗ 1 7→ (( u ⊗ u ) p G ( u ⊗ u ) ∗ ) ⊗ B . No w t ha t w e ha v e computed the pro duct δ ⊗ R U ( X,ϕ,P ) ∆ and established the notation in the sequel, we b egin the un t wisting step. In particular, w e shall deﬁne an automor- phism Θ : R S ( X , ϕ, Q ) ⊗ S → R S ( X , ϕ, Q ) ⊗ S whic h is homotopic to 1 R S ( X,ϕ,Q ) ⊗ S in K K ( R S ( X , ϕ, Q ) ⊗ S , R S ( X , ϕ, Q ) ⊗ S ) and therefore taking the in tersection pro duct of δ ⊗ R U ( X,ϕ,P ) ∆ with Θ do es not c ha nge the class. Indeed, for a ∈ S ( X, ϕ, Q ), u implem en t ing the action α s , and f ( t ) ∈ S = C 0 (0 , 1), deﬁne Θ ( a · u k ⊗ f ( t )) = a · u k ⊗ e 2 π ik t f ( t ) . K-THEORETIC DU ALITY FOR HYPERBOLIC DYNAMICAL SY STEMS 27 W e note that this map is give n on generators by 1 ⊗ z 7→ 1 ⊗ z , a ⊗ 1 7→ a ⊗ 1 , u ⊗ 1 7→ u ⊗ z . A t this p oint w e need to accomplish tw o things. First, w e must sho w that Θ is homotopic to 1 R S ( X,ϕ,Q ) ⊗ S and second, we must compute the pro duct Θ ⊗ ( R S ( X,ϕ,Q )) ⊗ S ( δ ⊗ R U ( X,ϕ,P ) ∆) . 7.2. Lemma. Ther e exists an automorphism Θ : R S ( X , ϕ, Q ) ⊗ S → R S ( X , ϕ, Q ) ⊗ S g iven by Θ ( a · u k ⊗ f ( t )) = a · u k ⊗ e 2 π ik t f ( t ) . Mor e ov er, Θ is the identity in K K ( R S ( X , ϕ, Q ) ⊗ S , R S ( X , ϕ, Q ) ⊗ S ) Pr o o f . W e will sho w a homotop y fr o m Θ to the iden tity . Indeed, for r ∈ [0 , 1 ] deﬁne, Θ r ( a · u k ⊗ f ( t )) = a · u k ⊗ e 2 π ik r t f ( t ) . W e w a nt to show that Θ r is a ∗ - automorphism f o r ev ery r in [0 , 1] and t in (0 , 1 ). T o see that co v ar ia nce is maintained, for all r in [0 , 1 ] and t in (0 , 1), w e compute Θ r ( u ⊗ 1) Θ r ( a ⊗ f ( t )) Θ r ( u ⊗ 1) ∗ = ( u ⊗ e 2 π ir t )( a ⊗ f ( t ))( u ∗ ⊗ e − 2 π ir t ) = uau ∗ ⊗ f ( t ) = α s ( a ) ⊗ f ( t ) = Θ r ( α s ( a ) ⊗ f ( t )) . Therefore, the map Θ r satisﬁes the cov ariance conditions for all r ∈ [0 , 1] a nd t in (0 , 1), so extends to a ∗ -homo mo r phism o n R S ( X , ϕ, Q ) ⊗ S . W e can explicitly write a fo rm ula fo r the inv erse of Θ r on g enerato r s so that Θ r is actually a ∗ -automorphism. Moreov er, Θ r is clearly is faithful since R S ( X , ϕ, Q ) ⊗ S is simple and Θ is clearly on to. No w w e mus t show that eac h map a · u k ⊗ f ( t ) 7→ Θ r ( a · u k ⊗ f ( t )) = a · u k ⊗ e 2 π ik tr f ( t ) is contin uous. Let ε > 0, and set δ = ε k M where M is the maxim um v alue of k a · u k ⊗ f ( t ) k for t ∈ (0 , 1). F or | r − r ′ | < δ w e compute k Θ r ( a · u k ⊗ f ( t )) − Θ r ′ ( a · u k ⊗ f ( t )) k = k a · u k ⊗ e 2 π ik tr f ( t ) − a · u k ⊗ e 2 π ik tr ′ f ( t ) k = k a · u k ⊗ (1 − e 2 π ik t ( r ′ − r ) ) f ( t ) k = | 1 − e 2 π ik t ( r ′ − r ) |k a · u k ⊗ f ( t ) k < ε M k a · u k ⊗ f ( t ) k ≤ ε. Finally , Θ 0 ( a · u k ⊗ f ( t )) = a · u k ⊗ e 2 π ik 0 t f ( t ) = a · u k ⊗ f ( t ) Θ 1 ( a · u k ⊗ f ( t )) = a · u k ⊗ e 2 π ik 1 t f ( t ) = Θ ( a · u k ⊗ f ( t )) so that Θ 0 = Id and Θ 1 = Θ . Therefore, Θ is homotopic to 1 R S ( X,ϕ,Q ) ⊗ S in K K ( R S ( X , ϕ, Q ) ⊗ S , R S ( X , ϕ, Q ) ⊗ S ).  7.3. Lemma. The class of Θ ⊗ R S ( X,ϕ,Q ) ⊗ S ( δ ⊗ R U ( X,ϕ,P ) ∆) in K K 1 ( R S ( X , ϕ, Q ) ⊗ S , R S ( X , ϕ, Q )) is r epr es ente d by the extension 0 ✲ R S ( X , ϕ, Q ) ⊗ K ( H ) ✲ E ′′ ✲ R S ( X , ϕ, Q ) ⊗ S ✲ 0 . F urthermor e, this extension r epr es ents the same class in K K - the o ry as δ ⊗ R U ( X,ϕ,P ) ∆ . 28 JER OME KAMINKER , IAN F. PUTNAM 1 , AND MICHAEL F. WH ITT AKER The pro of is analogo us to the pro of of Lemma 7.1 . Ho w ev er, we note that E ′′ is the C ∗ -algebra generated b y R S ( X , ϕ, Q ) ⊗ K ( H ) and the image of the map (1 ⊗ π u ⊗ π s ) ◦ ( δ ⊗ 1) ◦ σ ∗ ◦ Θ : R S ( X , ϕ, Q ) ⊗ S → B ( H ⊗ H ⊗ ℓ 2 ( Z )) . F urt hermore, the image of a · u k ⊗ z − 1 completely determines the ab o v e map, whic h is described on generators by 1 ⊗ z 7→ (( u ⊗ u ) p G v ∗ ) ⊗ 1 , (16) a ⊗ 1 7→ [(( u ⊗ u ) p G ( u ⊗ u ) ∗ ) ⊗ 1)][ U (1 ⊗ a ⊗ 1) U ∗ ] , (17) u ⊗ 1 7→ (( u ⊗ u ) p G v ∗ ) ⊗ 1 . (18) T o complete the pro of w e m ust sho w t hat the class Θ ⊗ R S ( X,ϕ,Q ) ⊗ S ( δ ⊗ R U ( X,ϕ,P ) ∆) in K K 1 ( R S ( X , ϕ, Q ) ⊗ S , R S ( X , ϕ, Q )) is equiv alen t to τ R S ( X,ϕ,Q ) ( T 0 ). Once w e hav e ac- complished this then Bott p erio dicit y implies t he result, see Section 4. W e b egin with a tec hnical construction to pro duce a unitary op erator. Supp ose that ( F , G ) is an ǫ ′ X -partition of X , as in section 5.6. W e deﬁne a vec tor, which we denote χ G in H = ℓ 2 ( X h ( P , Q )) whic h tak es the v alue (# G ) − 1 / 2 on the set G and zero elsewhere. Note that χ G is a unit vec tor. W e let q G denote the rank one pro jection on t o the span of χ G . W e deﬁne W G b y setting, fo r all y in X h ( P , Q ), W G ( δ y ⊗ χ G ) = X k f k ( y ) δ [ y ,g k ] ⊗ δ [ g k ,y ] , where the sum is tak en ov er all k suc h tha t y is in B ( g k , ε ′ X / 2). R ecall that in an ε ′ X - partition of X the supp ort of f k is con ta ined in B ( g k , ε ′ X / 2). Using our standard con v ention (the brack et returns zero if it is no t deﬁned), w e will simply write the sum ab ov e as b eing o v er all k = 1 , 2 , . . . , K since f k ( y ) will b e zero if [ y , g k ] and [ g k , y ] fail to b e deﬁned. W e also set W G ( δ z ⊗ ξ ) = 0, for ξ in H o rthogonal to χ G . It is easy to verify that W ∗ G ( δ y ⊗ δ z ) =  f k ([ y , z ]) δ [ y ,z ] ⊗ χ G if y ∈ X u ( g k , ε ) , z ∈ X s ( g k , ε ) 0 otherwise for all w , z in X h ( P , Q ). The following lemma summarizes the basic prop erties of W G . 7.4. Lemma. Supp ose that ( F , G ) is an ǫ ′ X -p artition of X and W G is deﬁne d as ab ove. Then (1) W ∗ G W G = 1 ⊗ q G . (2) W G W ∗ G = p G . (3) If ( F ◦ ϕ − 1 , ϕ ( G )) is also an ε ′ X -p artition, then ( u ⊗ u ) W G ( u ⊗ u ) ∗ = W ϕ ( G ) . (4) W G ( S ( X , ϕ , Q ) ⊗ K ) ⊂ S ( X , ϕ, Q ) ⊗ K . Pr o o f . The ﬁrst t hr ee it ems are the result of direct computations, whic h we o mit. F or t he fourth item, let a b e in S ( X , ϕ, Q ) and supp o se k is any compact op erator . No w W G ( a ⊗ k ) = W G (1 ⊗ q G )( a ⊗ k ) = W G ( a ⊗ q G )(1 ⊗ k ) , K-THEORETIC DU ALITY FOR HYPERBOLIC DYNAMICAL SY STEMS 29 and so it suﬃces to show W G ( a ⊗ q G ) is in S ( X , ϕ, Q ) ⊗ K . The C ∗ -algebra S ( X , ϕ, Q ) has an a pproximate iden tit y consisting of con tinuous functions of compact supp o rt o n X u ( Q ). Moreo ver, suc h functions are spanned by elemen ts supp orted on sets of the form X u ( v , ǫ ′ X / 2). So it suﬃces to cons ider a p oin t v in X u ( Q ), a function a in S ( X , ϕ, Q ) supp o rted on a basic set o f the form V s ( v , v, h s , ǫ ′ X / 2) suc h that aδ y = a ( y , y ) δ y if y is in X u ( v , ǫ ′ X / 2) and zero otherwise, and prov e that W G ( a ⊗ q G ) is in S ( X , ϕ, Q ) ⊗ K . F o r each k , deﬁne a function b k supp orted on a basic set of the form V s ([ v , g k ] , v , h s , ǫ ′ X / 2) b y b k ( y ′ , y ) = a ( y , y ) f k ( y ) if d ( y , g k ) < ǫ ′ X and [ y , g k ] = y ′ and to b e zero o therwise. Also deﬁne e k to b e the rank one op erator whic h maps χ G to δ [ g k ,v ] and is zero on the orthogonal complemen t of χ G . It fo llo ws that b k is in S ( X , ϕ, Q )and a computation shows that W G ( a ⊗ q G ) = X k b k ⊗ e k ∈ S ( X , ϕ, Q ) ⊗ K .  7.5. Lemma. L et a b e in S ( X , ϕ, Q ) an d le t ( F , G ) b e an ε ′ X -p artition. T hen we have lim n →∞ k (1 ⊗ α n s ( a )) W G − W G ( α n s ( a ) ⊗ 1) k = 0 . Pr o o f . It suﬃces to prov e the result fo r a supp orted in a basic set of the form V s ( v , w, h s , δ ) and further, since w e are taking limits as n go es to p ositiv e inﬁnit y , we ma y also assume that v and w are within ε ′ X / 2 so that h s is giv en by the brac ket map. W e observ e that b oth op erator s (1 ⊗ α n s ( a )) W G and W G (1 ⊗ α n s ( a )) are zero o n the or- thogonal compleme n t o f ℓ 2 ( X h ( P , Q )) ⊗ C · χ G . W e consider y in H = ℓ 2 ( X h ( P , Q )) and compute (1 ⊗ α n s ( a )) W G ( δ y ⊗ χ G ) = (1 ⊗ α n s ( a )) X k f k ( y ) δ [ y ,g k ] ⊗ δ [ g k ,y ] = X k f k ( y ) a ([ ϕ − n [ g k , y ] , v ] , ϕ − n [ g k , y ]) δ [ y ,g k ] ⊗ δ ϕ n [ ϕ − n [ g k ,y ] ,v ] and also W G ( α n s ( a ) ⊗ 1)( δ y ⊗ χ G ) = W G a ([ ϕ − n ( y ) , v ] , ϕ − n ( y )) δ ϕ n [ ϕ − n ( y ) ,v ] ⊗ χ P = X k f k ( ϕ n [ ϕ − n ( y ) , v ]) a ([ ϕ − n ( y ) , v ] , ϕ − n ( y )) δ [ ϕ n [ ϕ − n ( y ) ,v ] ,g k ] ⊗ δ [ g k ,ϕ n [ ϕ − n ( y ) ,v ]] . Let ε > 0 b e giv en. Let M b e an upp er b ound o n the function | a | . W e may ﬁnd a constan t ε 1 > 0 suc h that | f k ( y ) − f k ( z ) | < ǫ/ 2 M K , for all y , z with d ( y , z ) < ǫ 1 and 1 ≤ k ≤ K . In addition, we se lect ǫ 1 > 0 suc h that | a ([ y , v ] , y ) − a ([ z , v ] , z ) | < ǫ/ 2 K for all y , z with z in X u ( y , ǫ 1 ). W e choo se N suﬃcien tly large so that λ − n ǫ X / 2 < ǫ 1 and λ − n ǫ X < ǫ ′ X , for all n ≥ N . With n ≥ N , holding k ﬁxed fo r the moment, we make the claim that if the co eﬃcien t in either expression ab o v e: f k ( y ) a ([ ϕ − n [ g k , y ] , v ] , ϕ − n [ g k , y ]) or f k ( ϕ n [ ϕ − n ( y ) , v ]) a ([ ϕ − n ( y ) , v ] , ϕ − n ( y )) is not zero, then w e hav e (1) [ y , g k ] = [ ϕ n [ ϕ − n ( y ) , v ] , g k ], (2) ϕ n [ ϕ − n [ g k , y ] , v ] = [ g k , ϕ n [ ϕ − n ( y ) , v ]], 30 JER OME KAMINKER , IAN F. PUTNAM 1 , AND MICHAEL F. WH ITT AKER (3) the map sending ( y , g k ) to ([ y , g k ] , ϕ n [ ϕ − n [ g k , y ] , v ]) is inj ective , (4) ϕ − n [ g k , y ] is in X u ( ϕ − n ( y ) , ǫ 1 ), (5) d ( ϕ n [ ϕ − n ( y ) , v ] , y ) < ǫ 1 . If f k ( y ) a ([ ϕ − n [ g k , y ] , v ] , ϕ − n [ g k , y ]) is no n- zero, then f k ( y ) m ust b e non- zero and this means that y is in B ( g k , ǫ ′ X / 2). Moreo v er, from the c hoice of ǫ ′ X , w e hav e that [ g k , y ] is in X u ( y , ǫ X / 2) and hence ϕ − n ([ g k , y ]) is in X u ( ϕ − n ( y ) λ − n ǫ X / 2). In addition, w e kno w that a ([ ϕ − n [ g k , y ] , v ] , ϕ − n [ g k , y ]) is non-zero and this means that ϕ − n ([ g k , y ]) is in X u ( w , ǫ ′ X / 2) and it follows that ϕ − n ( y ) is in X u ( w , ǫ ′ X / 2 + λ − n ǫ X / 2). Since λ − n ǫ X < ǫ ′ X < ǫ X / 2, w e see that [ ϕ − n ( y ) , v ] is also deﬁned and is in X s ( ϕ − n ( y ) , ǫ X / 2). It follows that ϕ n [ ϕ − n ( y ) , v ] is in X s ( y , λ − n ǫ X / 2) and a lso in B ( g k , ǫ ′ X ). If the se cond expression, f k ( ϕ n [ ϕ − n ( y ) , v ]) a ([ ϕ − n ( y ) , v ] , ϕ − n ( y )) is non-zero, then w e mu st ha ve tha t a ([ ϕ − n ( y ) , v ] , ϕ − n ( y )) is non-zero and so ϕ − n ( y ) is in X u ( w , ǫ ′ X / 2). Then we ha ve [ ϕ − n ( y ) , v ] is in X s ( ϕ − n ( y ) , ǫ X / 2) and hence ϕ n [ ϕ − n ( y ) , v ] is in X s ( y , λ − n ǫ X / 2). In addition, if the co eﬃcien t is non-zero, the f k term is non-zero and t his means that this same point is in B ( g k , ǫ ′ X / 2) and hence, y is in B ( g k , ǫ ′ X / 2 + λ − n ǫ X / 2). Since λ − n ǫ X < ǫ ′ X , [ g k , y ] is in X u ( y , ǫ X / 2). T o summarize, if either term is non-zero, then w e ha v e [ g k , y ] is deﬁned and is in X u ( y , ǫ X / 2), ϕ − n ( y ) is in X u ( w , ǫ ′ X ) and [ ϕ − n ( y ) , v ] is deﬁned and in X s ( ϕ − n ( y ) , ǫ X / 2). Parts 4 and 5 of the claim follow at once since λ − n ǫ X / 2 < ǫ 1 . F o r any 0 ≤ m ≤ n , w e hav e d ( ϕ m − n ( y ) , ϕ m [ ϕ − n ( y ) , x ]) ≤ λ − m d ( ϕ − n ( y ) , [ ϕ − n ( y ) , v ]) ≤ λ − m ǫ X / 2 ≤ ǫ X / 2 , and d ( ϕ m − n ( y ) , ϕ m − n [ g k , y ]) ≤ λ − n + m d ( y , [ g k , y ]) ≤ λ − n + m ǫ X / 2 ≤ ǫ X / 2 . F r o m the triangle inequality , we ha ve d ( ϕ m [ ϕ − n ( y ) , v ] , ϕ m − n [ g k , y ]) ≤ ǫ X . This means that the brac k et of these points is deﬁned (in either or der). First, taking brac k et in the order giv en and using the ϕ -inv ariance of the br a c ke t w e hav e [ ϕ m [ ϕ − n ( y ) , v ] , ϕ m − n [ g k , y ]] = ϕ m [[ ϕ − n ( y ) , v ] , ϕ − n [ g k , y ]] = ϕ m [ ϕ − n ( y ) , ϕ − n [ g k , y ]] . When m = n , the left hand side b ecomes [ ϕ n [ ϕ − n ( y ) , v ] , ϕ n − n [ g k , y ]] = [ ϕ n [ ϕ − n ( y ) , v ] , [ g k , y ]] = [ ϕ n [ ϕ − n ( y ) , v ] , y ]] while the rig h t ha nd side is ϕ n [ ϕ − n ( y ) , ϕ − n [ g k , y ]] = ϕ n ( ϕ − n ( y )) = y as ϕ − n [ g k , y ] is in X u ( ϕ − n ( y ) , λ − n ǫ X / 2). Now brack eting each with g k yields [ y , g k ] = [[ ϕ n [ ϕ − n ( y ) , v ] , y ] , g k ] = [ ϕ n [ ϕ − n ( y ) , v ] , g k ] and w e hav e established part 1 of the claim. On the other hand, if we bra ck et in the other order, and again use the ϕ - in v ariance, w e obtain [ ϕ m − n [ g k , y ] , ϕ m [ ϕ − n ( y ) , v ]] = ϕ m [ ϕ − n [ g k , y ] , [ ϕ − n ( y ) , v ]] = ϕ m [ ϕ − n [ g k , y ] , v ] . K-THEORETIC DU ALITY FOR HYPERBOLIC DYNAMICAL SY STEMS 31 Setting m = n , the left hand side is [ ϕ n − n [ g k , y ] , ϕ n [ ϕ − n ( y ) , v ]] = [[ g k , y ] , ϕ n [ ϕ − n ( y ) , v ]] = [ g k , ϕ n [ ϕ − n ( y ) , v ] while the rig h t is ϕ n [ ϕ − n [ g k , y ] , ϕ − n ( y )] . W e ha ve established the second part of t he claim. F o r the third part of the claim, let x = [ y , g k ] and z = [ g k , ϕ n [ ϕ − n ( y ) , v ]]. W e can recov er y and g k from x a nd z b y observing that [ z , x ] = g k and [ x, z ] = [[ y , g k ] , [ g k , ϕ n [ ϕ − n ( y ) , v ]]] = [ y , ϕ n [ ϕ − n ( y ) , v ]] = y , since ϕ n [ ϕ − n ( y ) , v ] is in X s ( y , ǫ ). W e ma y conclude from the ﬁrst t w o pa rts of our claim that (1 ⊗ α n s ( a )) W G − W G ( α n s ( a ) ⊗ 1)( δ y ⊗ χ G ) = X k ( f k ( y ) a ([ ϕ − n [ g k , y ] , v ] , ϕ − n [ g k , y ]) − f k ( ϕ n [ ϕ − n ( y ) , v ]) a ([ ϕ − n ( y ) , v ] , ϕ − n ( y ))) δ [ y ,g k ] ⊗ δ [ g k ,ϕ n [ ϕ − n ( y ) ,v ]] . F r o m part 3, w e see that the v ectors app earing in the right hand side of the expression at the end o f the last paragraph are pairwise orthogonal and the sums obta ined for diﬀeren t v alues o f y are pairwise orthogonal. F rom this it follow s that k (1 ⊗ α n s ( a )) W G − W G ( α n s ( a ) ⊗ 1) k 2 = sup y ∈ X h ( P, Q ) X k | f k ( y ) a ([ ϕ − n [ g k , y ] , v ] , ϕ − n [ g k , y ]) − f k ( ϕ n [ ϕ − n ( y ) , v ]) a ([ ϕ − n ( y ) , v ] , ϕ − n ( y )) | 2 . The estimate that, for ﬁxed k and y , | f k ( y ) a ([ ϕ − n [ g k , y ] , v ] , ϕ − n [ g k , y ]) − f k ( ϕ n [ ϕ − n ( y ) , v ]) a ([ ϕ − n ( y ) , v ] , ϕ − n ( y )) | < ǫ/K follo ws from the last tw o parts of our claim and standard tec hniques. This completes the pro of.  W e next deﬁne d W G =  W G (1 − W G W ∗ G ) 1 / 2 − (1 − W ∗ G W G ) 1 / 2 W ∗ G  whic h is a unitary op erator. Moreo ver, it follows f rom part 4 of Lemma 7.4 that it is in the m ultiplier algebra of S ( X , ϕ, Q ) ⊗ K . 7.6. Lemma. L et a b e in S ( X , ϕ, Q ) an d le t ( F , G ) b e an ε -p artition. Th en we have lim n → + ∞ k (( p G (1 ⊗ α n ( a ))) ⊗ e 1 , 1 ) d W G − d W G (( α n ( a ) ⊗ q G ) ⊗ e 1 , 1 ) k = 0 . Mor e ov er, if ( F ◦ ϕ − 1 , ϕ ( G )) is also an ε ′ X -p artition, then \ W ϕ ( G ) is also in the multiplier algebr a of S ( X , ϕ, Q ) ⊗ K and the analo gous r esult holds with \ W ϕ ( G ) , p ϕ ( G ) , and q ϕ ( G ) . 32 JER OME KAMINKER , IAN F. PUTNAM 1 , AND MICHAEL F. WH ITT AKER Pr o o f . First, notice that p G is in S ⊗ U , while α n ( a ) is in S . It follows from Lemma 6.3 tha t in the ﬁrst term w e hav e lim n →∞ k p G (1 ⊗ α n s ( a )) − (1 ⊗ α n s ( a )) p G k = 0 . So it suﬃces to prov e the result a fter interc hang ing the order of p G and 1 ⊗ α n s ( a ). It follo ws fro m the fact that W G W ∗ G = p G that the (new) ﬁrst term a b o v e is (((1 ⊗ α n s ( a )) p G ) ⊗ e 1 , 1 ) d W G = ((1 ⊗ α n s ( a )) ⊗ e 1 , 1 )( p G ⊗ e 1 , 1 ) d W G = ((1 ⊗ α n s ( a )) ⊗ e 1 , 1 )( W G ⊗ e 1 , 1 ) = (1 ⊗ α n s ( a )) W G ⊗ e 1 , 1 . On the o t her hand, using the fact that W ∗ G W G = 1 ⊗ q G , the second term ab ov e is d W G (( α n s ( a ) ⊗ q G ) ⊗ e 1 , 1 ) = d W G ((1 ⊗ q G ) ⊗ e 1 , 1 )( α n s ( a ) ⊗ 1) ⊗ e 1 , 1 ) = ( W G ⊗ e 1 , 1 )(( α n s ( a ) ⊗ 1) ⊗ e 1 , 1 ) = W G (( α n s ( a ) ⊗ 1) ⊗ e 1 , 1 ) . The ﬁrst statemen t now follow s at o nce from Lemma 7.5 and the second statemen t follows from com bining the ﬁrst statemen t with part 3 o f lemma 7 .4.  Let us denote t he map (1 ⊗ π u ⊗ π s ) ◦ ( δ ⊗ 1) ◦ σ ∗ ◦ Θ : R S ( X , ϕ, Q ) ⊗ S → B ( H ⊗ H ⊗ ℓ 2 ( Z )) b y ψ . Observ e that ψ ( a · u k ⊗ z l − 1) determines the extension 0 ✲ R S ( X , ϕ, Q ) ⊗ K ( H ) ✲ E ′′ ✲ R S ( X , ϕ, Q ) ⊗ S ✲ 0 , represen ting the class Θ ⊗ R S ( X,ϕ,Q ) ⊗ S ( δ ⊗ R U ( X,ϕ,P ) ∆) in K K 1 ( R S ( X , ϕ, Q ) ⊗ S , R S ( X , ϕ, Q )). Therefore, using Lemma 7.6 and equiv alence in K K - theory , w e ha v e ψ ( a · u k ⊗ z l − 1 ) = ( \ W ϕ ( G ) ⊗ 1 ) ∗ ( ψ ( a · u k ⊗ z l − 1 ) ⊗ e 1 , 1 )( \ W ϕ ( G ) ⊗ 1) = ( W ϕ ( G ) ⊗ 1 ) ∗ ( ψ ( a · u k ⊗ z l − 1 ))( W ϕ ( G ) ⊗ 1 ) . No w, com bining Lemma 6.1, Lemma 6.2, and Lemma 7.6, w e compute ( W ∗ ϕ ( G ) ⊗ 1) U (1 ⊗ a ⊗ 1) U ∗ ( W ϕ ( G ) ⊗ 1 ) =  U ( a ⊗ 1 ⊗ 1) U ∗ (1 ⊗ q G ⊗ 1 ) if n ≥ 0 0 if n < 0 where equality is up to the ideal R S ( X , ϕ, Q ) ⊗ K ( H ). T o further simplify notatio n, let us also deﬁne U = W ∗ ϕ ( G ) (( u ⊗ u ) p G v ∗ ) W ϕ ( G ) ∈ B ( H ⊗ H ) . Using the notat io n and computations from the preceding parag r a ph tog ether with the maps describ ed in (16)-(1 8) on page 28, w e o btain, for a in S ( X , ϕ, Q ), ψ ( a · u k ⊗ z l − 1) = ( U ⊗ B ∗ ) l + k U ( a ⊗ 1 ⊗ 1) U ∗ (1 ⊗ B ) k − ( U ⊗ B ∗ ) k U ( a ⊗ 1 ⊗ 1) U ∗ (1 ⊗ B ) k as a b ounded op erator on the Hilb ert space H ⊗ q G ⊗ ℓ 2 ( N ), where w e hav e replaced ℓ 2 ( Z ) b y ℓ 2 ( N ) since the op erato r ψ ( a · u k ⊗ z l − 1) is zero on the subspace { 1 ⊗ q G ⊗ δ n | n < 0 } . Therefore, note that B is a one sided shift on ℓ 2 ( N ). K-THEORETIC DU ALITY FOR HYPERBOLIC DYNAMICAL SY STEMS 33 W e are left to show that w e hav e the following isomorphism of extensions, where E ′′′ is the C ∗ -algebra generated b y the image of ψ and the ideal R S ( X , ϕ, Q ) ⊗ K ( H ⊗ ℓ 2 ( N )), 0 ✲ R S ( X, ϕ, Q ) ⊗ q G ⊗ K ( ℓ 2 ( N )) ✲ E ′′′ π ✲ R S ( X, ϕ, Q ) ⊗ S ✲ 0 0 ✲ R S ( X, ϕ, Q ) ⊗ K ( ℓ 2 ( N )) ∼ = ❄ ✲ R S ( X, ϕ, Q ) ⊗ C ∗ ( B − 1) β ❄ ✲ R S ( X, ϕ, Q ) ⊗ S w w w w w w w w w ✲ 0 . Indeed, the quotient map π : E ′′′ → R S ( X , ϕ, Q ) ⊗ S is giv en on g enerato r s b y U ⊗ 1 7→ u ⊗ 1 , 1 ⊗ B 7→ u ⊗ z , U ( a ⊗ 1 ⊗ 1) U ∗ 7→ a ⊗ 1 and the map β is giv en on generators b y U ⊗ 1 7→ u ⊗ 1 , 1 ⊗ B 7→ u ⊗ B , U ( a ⊗ 1 ⊗ 1) U ∗ 7→ a ⊗ 1 . The reader is left to sho w that β is an isomorphism and that the ab ov e diagram com- m utes. The second extension represen ts the class τ R S ( X,ϕ,Q ) ( T 0 ), whic h is K K - equiv alen t to 1 R S ( X,ϕ,Q ) b y Bott p erio dicit y . This completes the pro of. 8. Concluding re marks and ques tions 8.1. Existence of the dualit y classes. As w e saw in Section 5, the construction of the K- theory dualit y elemen t did not require t he expanding and contracting nature of the dynamics. An essen tial pro p ert y w as that the stable and unstable relat io ns in tersected at a coun ta ble set of p o in ts. Recall that a transv ersal to a foliation is a set whic h meets eac h leaf in a coun table set, so the condition that eac h stable equiv alence class meets each unstable class in a coun table set is a tr a nsv ersalit y condition. Based on the example of transv erse folia tions, a notio n of t r ansv erse group oids has b een suggested and it is hop ed that the axioms will b e suﬃcien t for the construction of a K - theory duality class in K K i ( C ∗ ( G 1 ) , C ∗ ( G 2 )) when G 1 and G 2 are transv erse group oids. On the other hand, the existenc e of the class ∆ requires h yp erb olicit y . This is what might b e exp ected from the Dirac-dual D irac appro ac h to the Baum-Connes conjecture. In that case, the construction of the dual D irac elemen t , whic h is the analog of our ∆, requires the use of a non-p ositive ly curv ed space. F or suc h a space the geo desic ﬂow will b e h yp erb olic. It would b e in teresting to ﬁnd fundamen tally diﬀerent types of conditions whic h would lead to a K- homology dualit y class, or to understand why there no o t her p ossibilities. 8.2. Crossed products and dynam ics. The relation b etw een a hyperb olic group acting (amenably) on its Gro mo v b oundar y and hyperb olic dynamical systems is an in teresting sub ject. One migh t ho p e that in general o ne could do as Bo wen and Series did, and reco de the action of certain F uc hsian g roups on their b oundaries to obt a in a single transformation with an a sso ciated Mark ov pa r t ition, hence one gets a subshift of ﬁnite type. Spielb erg sho we d that the Ruelle algebras fo r the subshift are isomorphic to crossed pro duct a lg ebras. This w as extended by Laca-Spielb erg and Delaro c he, but in those cases it was necess ary to use the f act that the a lgebras satisﬁed the h yp othesis of the Phillips-Kirc hberg theorem, and sho w that their K - theories w ere the same, to deduce that they were isomorphic. This suggest that the appropriat e setting for relating amenable actions of hyperb olic g r oups on 34 JER OME KAMINKER , IAN F. PUTNAM 1 , AND MICHAEL F. WH ITT AKER their b oundaries to hy p erb olic dynamics is via identifying the crossed pro duct C ∗ -algebras with Ruelle algebras of Smale spaces. 8.3. Ruelle algebras as build ing blo c ks for algebras asso ciated to diﬀeom orphisms of manifolds. Let f : M → M be an Axiom A diﬀeomorphism of a compact manifold, as deﬁned by Smale [40]. 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Dep ar tment of Ma thema tical Sciences, IUPUI , Indianapolis, IN Curr ent addr ess : Depar tment of Mathematics, UC Da vis , Davis, CA 95616 E-mail addr ess : kam inker @math. ucdavis.edu Dep ar tment of Ma thema tics and St a tistics, University o f Victoria, V ictoria, B. C., Canada V8W 3R4 E-mail addr ess : put nam@m ath.uv ic.ca Dep ar tment of Ma thema tics and St a tistics, University o f Victoria, V ictoria, B. C., Canada V8W 3R4 E-mail addr ess : mfw hitta ker@gm ail.com Curr ent addr ess : Depar tment of Mathematics, W ollongo ng Univ er sity , Australia

K-Theoretic Duality for Hyperbolic Dynamical Systems

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