Spectral triples for hyperbolic dynamical systems
aa r X i v : . [ m a t h . OA ] A ug SPECTRAL TRIPLES FOR HYPERBOLIC DYNAMICAL SYSTEMS
MICHAEL F. WHITTAKER
Abstract.
Spectral triples are defined for C ∗ -algebras associated with hyperbolic dynam-ical systems known as Smale spaces. The spectral dimension of one of these spectral triplesis shown to recover the topological entropy of the Smale space. Introduction
A spectral triple ( A, H , D ) consists of a faithful representation of a C ∗ -algebra A asbounded operators on a separable Hilbert space H along with a self-adjoint, unboundedoperator on H satisfying the additional conditions(1) the set { a ∈ A | [ D, a ] ∈ B ( H ) } is norm dense in A and(2) the operator a (1 + D ) − is a compact operator on H for all a in A .Alain Connes developed spectral triples as a generalization of a Fredholm module whichputs the spectrum of an unbounded, self-adjoint operator at the forefront [4]. Using spectraltriples Connes was able to recover geometric data from commutative algebras in a frameworkthat extended to the noncommutative case [4, 5]. By now spectral triples are at the forefrontin Connes’ noncommutative geometry and play a key role in the noncommutative analogueof the calculus. In this paper, we investigate spectral triples for C ∗ -algebras associatedwith hyperbolic dynamical systems known as Smale spaces. These are the C ∗ -algebrasintroduced by David Ruelle in his investigation of Gibbs states associated with hyperbolicdiffeomorphisms, which are of the type introduced by Connes in connection with foliations[21].Let us begin with a heuristic definition of a Smale Space. Suppose ( X, d ) is a compactmetric space and ϕ : X → X is a homeomorphism. We say ( X, d, ϕ ) is a Smale Space if X islocally a hyperbolic product space with respect to ϕ ; that is, there is a global constant ε X > x is in X we have two sets X s ( x, ε X ) and X u ( x, ε X ) whose intersection is { x } and the Cartesian product of these sets is homeomorphic to a neighborhood of x . Moreover,for any points y and z in X s ( x, ε X ) we require that d ( ϕ ( y ) , ϕ ( z )) < λ − d ( y, z ) where λ > X u ( x, ε X ) has the same property if we replace ϕ with ϕ − . Wecall X s ( x, ε X ) and X u ( x, ε X ) the local stable and unstable sets of x respectively.David Ruelle introduced Smale spaces as a purely topological description of the basicsets of Axiom A diffeomorphisms on a compact manifold [20]. A basic set is a closed, ϕ -invariant subset of the manifold but does not need to be a manifold itself. In fact, thesesets are usually fractal and have no smooth structure whatsoever. We note that, under mildconditions, Smale spaces are chaotic dynamical systems. We also remark that examples of Date : December 26, 2017.2010
Mathematics Subject Classification.
Primary 47C15; Secondary 37D20,47B25.Research supported in part by ARC grant 228-37-1021, Australia.
Smale spaces include shifts of finite type, solenoids, the dynamical systems associated withcertain substitution tilings, and hyperbolic toral automorphisms.Several C ∗ -algebras can be associated with a Smale space. The first algebra we wishto study is the C ∗ -algebra associated with the stable equivalence relation. To do so, itis most convenient to find a transversal, so that it becomes an ´etale equivalence relation.Natural transversals are available as the unstable equivalence classes, but care must be usedwhen defining a suitable topology on the groupoid of stable equivalence restricted to theunstable transversal. The situation is simplified when the transversal is ϕ -invariant, so thetransversal is defined to be the unstable equivalence classes of a ϕ -invariant set of periodicpoints. A groupoid C ∗ -algebra is produced which first appeared in [16] and is strongly Moritaequivalent to the stable C ∗ -algebra appearing in [14]. The unstable C ∗ -algebra is constructedin an analogous fashion. Furthermore, the homeomorphism ϕ gives rise to an automorphismon both the stable and unstable algebras and the crossed products are known as the stableand unstable Ruelle algebras [14]. We shall define spectral triples on all of these C ∗ -algebras.To define a spectral triple we begin by considering specific classes of the stable (unstable)equivalence relation. In our situation, each equivalence class can be associated with a point inthe orbit of a periodic point. These periodic orbits are viewed as attractors in the sense that,given ε > x in an equivalence class associated with a periodic point, thereis an integer N such that the distance between the periodic orbit and ϕ n ( x ) is within ε forall n ≥ N . A similar result is true on the unstable equivalence relation provided we replace ϕ with ϕ − . A function is defined on the equivalence classes of an orbit which essentiallycounts the number of iterations of ϕ required to move each point into a fixed neighbourhoodof the orbit of the associated periodic point. Moreover, if the point begins in this fixedneighbourhood then the function will count the number of inverse iterations required toremove the point from the neighbourhood. Using this function, we define a Dirac operator D , which gives rise to a spectral triple on the stable algebra. A similar construction definesa spectral triple on the unstable algebra. Furthermore, the Dirac operator D commuteswith the automorphism used to define the crossed product Ruelle algebras and therefore thespectral triple defined extends to the Ruelle algebras as well. All of these spectral triplesturn out to be θ -summable; that is, the operator e − (1+ D ) is trace class.A much more desirable property for spectral triples is finite summability. A spectral triple( A, H , D ) is finitely summable when the operator (1 + D ) − p/ is trace class for some p ∈ R .The infimum over all such p ∈ R is called the spectral dimension of the spectral triple.Defining a new Dirac operator by D = λ D , where λ > D is the aforementioned Dirac operator, we obtain a finitely summablespectral triple provided we make certain assumptions on the function used to define D . Wenote that this spectral triple does not extend to the Ruelle algebras. Awknowledgements.
Great acclamation is due to Ian Putnam who supervised my work duringmy doctoral studies, from which this note is based.2.
Smale Spaces
In the introduction, we gave a heuristic definition of a Smale space and in this section wecomment on how to make this definition rigorous, as well as discussing properties required
PECTRAL TRIPLES FOR HYPERBOLIC DYNAMICAL SYSTEMS 3 in the sequel. The reader is encouraged to reference [14] and [20] for additional details onthese remarkable spaces.To make the definition of a Smale space rigorous requires us to postulate the existenceof constants ε X > λ > ε X > λ > d ( x, y ) < ε X , then { [ x, y ] } = X s ( x, ε X ) ∩ X u ( y, ε X ).The local stable and unstable sets of a point x in X are now defined by X s ( x, ε ) = { y ∈ X | d ( x, y ) < ε and [ y, x ] = x } and X u ( x, ε ) = { y ∈ X | d ( x, y ) < ε and [ x, y ] = x } where 0 ≤ ε ≤ ε X . Figure 1 illustrates the bracket with respect to these sets. 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 bracket map
Definition 2.1.
A dynamical system (
X, d, ϕ ) having a bracket map is a
Smale space .Moreover, a Smale space is said to be irreducible if the set of periodic points under ϕ aredense and there is a dense ϕ -orbit.There are canonical global stable and unstable equivalence relations on X . Given a point x in X we define the stable and unstable equivalence 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 } . We shall also employ the notation x ∼ s y if y is in X s ( x ) and x ∼ u y if y is in X u ( x ). Tosee the connection between the global stable and local stable set of a point, we note that,for any x in X and ε >
0, we have X s ( x, ε ) ⊂ X s ( x ). Furthermore, a point y is in X s ( x )if and only if there exists N ≥ ϕ n ( y ) is in X s ( ϕ n ( x ) , ε ) for all n ≥ N . Thisnontrivial fact follows from the expansive nature of ϕ in the unstable direction and is mosteasily observed when x is a fixed point. Indeed, since y ∈ X s ( x ), there exists N ∈ N such MICHAEL F. WHITTAKER that d ( ϕ n ( y ) , x ) < ε for all n ≥ N . Suppose that ϕ n ( y ) is not in the local stable set of x forsome n ≥ N ; that is, [ ϕ n ( y ) , x ] = x . By the definition of the bracket [ ϕ n ( y ) , x ] ∈ X u ( x, ε )and it follows that we can find m such that d ( ϕ m [ ϕ n ( y ) , x ] , x ) > ε and consequently that d ( ϕ m + n ( y ) , x ) > ε , a contradiction. A slightly more complex argument holds when x is nota fixed point. Making the obvious modifications, the same is true in the unstable situation.As topological spaces the stable and unstable equivalence classes are quite unseemly withrespect to the relative topology of X . In fact, if ( X, d, ϕ ) is irreducible it follows that boththe stable and unstable equivalence classes of orbits are dense in X [20]. To rectify thissituation we observe that the local stable sets form a neighborhood base for a topology onthe global stable sets; that is, given an equivalence class X s ( x ), the collection { X s ( y, δ ) | y ∈ X s ( x ) and δ > } is a neighbourhood base for a Hausdorff and locally compact topology on X s ( x ). We define a topology on the unstable equivalence classes in an analogous fashion.3. C ∗ -algebras of Smale Spaces In this Section we will construct C ∗ -algebras from an irreducible Smale space. These C ∗ -algebras are referred to as the stable and unstable algebras. In [21], David Ruelle constructed C ∗ -algebras from the stable and unstable equivalence relations. Putnam and Spielberg thenrefined these constructions in [16] and defined groupoids that are equivalent, in the sense ofMuhly, Renault, and Williams [12], to the stable and unstable groupoids, but which are ´etale.We follow the development in [16] and the reader is referred there for further properties ofthese algebras.The astute reader will have noticed that exchanging the homeomorphism ϕ with ϕ − interchanges the stable and unstable equivalence relations. This phenomenon persists at thelevel of the stable and unstable C ∗ -algebras as well. For this reason we omit any discussionof the unstable C ∗ -algebra since we can define the unstable algebra to be the stable algebraof the Smale space with ϕ exchanged with ϕ − .3.1. ´Etale groupoids on Smale Spaces. Let (
X, d, ϕ ) be a Smale space and let P and Q be finite sets of ϕ -invariant periodic points. At this point we make no restrictions on thesets P and Q , however, in the following section we will add the assumption that P and Q are disjoint. Define X s ( P ) = [ p ∈ P X s ( p ) , X u ( Q ) = [ q ∈ Q X u ( q ) , X h ( P, Q ) = X s ( P ) ∩ X u ( Q ) . Lemma 3.1 ([20]) . If ( X, d, ϕ ) is an irreducible Smale space and P and Q are both ϕ -invariant sets of periodic points, then X h ( P, Q ) is dense in X . Moreover, if P ∩ Q = ∅ , then X h ( P, Q ) does not contain any periodic points. We now define a groupoid on (
X, d, ϕ ) by G s ( X, ϕ, Q ) = { ( v, w ) | v ∼ s w and v, w ∈ X u ( Q ) } . We remark that G s ( X, ϕ, Q ) is a closed transversal to stable equivalence on (
X, d, ϕ ) in thesense of Muhly, Renault, and Williams [12].
PECTRAL TRIPLES FOR HYPERBOLIC DYNAMICAL SYSTEMS 5
We aim to define an ´etale topology G s ( X, ϕ, Q ). Suppose v ∼ s w and v, w ∈ X u ( Q ). Since v ∼ s w it follows that there exists N such that ϕ N ( w ) ∈ X s ( ϕ N ( v ) , ε X / , see Section 2. By the continuity of ϕ , define 0 < δ < ε X / ϕ n ( X u ( w, δ )) ⊂ X u ( ϕ n ( w ) , ε X /
2) for all 0 ≤ n ≤ N. Given
N, δ , we may now define a map h s on X u ( w, δ ) via h s ( x ) = ϕ − N [ ϕ N ( x ) , ϕ N ( v )] . Let δ ′ = sup { d ( v, h s ( x )) | x ∈ X u ( w, δ ) } . It is shown in [14] that the map h s : X u ( w, δ ) → X u ( v, δ ′ ) is a local homeomorphism. An illustration of the map h s is given in Figure 2. X s ( w ) X u ( w, δ ) wxX s ( v ) X u ( v, δ ′ ) vh s ( x ) = φ − N [ φ N ( x ) , φ N ( v )] X u ( ϕ N ( w ) , ε X / X u ( ϕ N ( v ) , ε X / φ N ( w ) φ N ( x ) φ N ( v )[ φ N ( x ) , φ N ( v )] φ N φ − N Figure 2.
The local homeomorphism h s : X u ( w, δ ) → X u ( v, δ ′ ) Lemma 3.2 ([14]) . Let v, w in X be such that v ∼ s w and v, w ∈ X u ( Q ) . There exists < δ, δ ′ ≤ ε X / and an integer N such that the map h s : X u ( w, δ ) → X u ( v, δ ′ ) is a localhomeomorphism. Theorem 3.3 ([14]) . Let v, w in X be such that v ∼ s w and v, w ∈ X u ( Q ) and let N , δ , δ ′ ,and h s be defined by lemma 3.2. The collection of sets V s ( v, w, h s , δ ) = { ( h s ( x ) , x ) | x ∈ X u ( w, δ ) , h s ( x ) ∈ X u ( v, δ ′ ) } MICHAEL F. WHITTAKER form a neighbourhood base for a topology on G s ( X, ϕ, Q ) . In this topology, the range andsource maps take each element in the neighbourhood base homeomorphically to an open setin X u ( Q ) . Moreover, this topology makes G s ( X, ϕ, Q ) a second countable, locally compact,Hausdorff groupoid. That is, G s ( X, ϕ, Q ) is an ´ etale groupoid. The Stable C ∗ -algebra of a Smale Space. We aim to study the groupoid C ∗ -algebraof the ´etale groupoid G s ( X, ϕ, Q ). To accomplish this, we apply Renault’s construction [17].Let C c ( G s ( X, ϕ, Q )) denote the continuous functions of compact support on G s ( X, ϕ, Q ),which is a complex linear space. A product and involution are defined on 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 ( x,z ) ∈ G s ( X,ϕ,Q ) f ( x, z ) g ( z, y ) f ∗ ( x, y ) = f ( y, x ) . This makes C c ( G s ( X, ϕ, Q )) into a complex ∗ -algebra.We aim to define a norm on C c ( G s ( X, ϕ, Q )) and then complete C c ( G s ( X, ϕ, Q )) in thisnorm to define a C ∗ -algebra. At this point there are several options. First we could lookat all possible representations of C c ( G s ( X, ϕ, Q )) as operators on a Hilbert space. Fromthese Hilbert spaces we obtain a norm and the completion is called the full C ∗ -algebra.Alternatively, we could consider a single representation on each equivalence class, called theregular representation. This gives rise to the reduced norm and the completion is the reduced C ∗ -algebra. In fact, it is shown in [16] that the groupoid of stable equivalence is amenableso that the full and reduced groupoid C ∗ -algebras are isomorphic. Definition 3.4.
The stable C ∗ -algebra, S ( X, ϕ, Q ), is the completion of C c ( G s ( X, ϕ, Q )) inthe reduced norm. When no confusion will arise S ( X, ϕ, Q ) is denoted by S .A third option is possible when ( X, d, ϕ ) is irreducible, which is called the fundamentalrepresentation [9, 16]. We aim to represent C c ( G s ( X, ϕ, Q )) as operators on the Hilbertspace H = ℓ ( X h ( P, Q )) . To that end, for f ∈ C c ( G s ( X, ϕ, Q )) and ξ ∈ H , define a representation π : C c ( G s ( X, ϕ, Q )) → B ( H ) via π ( f ) ξ ( x ) = X ( x,y ) ∈ G s ( X,ϕ,Q ) f ( x, y ) ξ ( y ) . With this formula, π ( f ) is a bounded linear operator on H . Moreover, we can complete π ( C c ( G s ( X, ϕ, Q ))) in the operator norm on this Hilbert space to obtain a C ∗ -algebra.Let us comment on the generality of this construction. In the case that ( X, d, ϕ ) is mixing,every stable and unstable equivalence class is dense. Moreover, X s ( P ) ∩ X u ( Q ) is densein X u ( Q ) so that π is a faithful representation to the reduced C ∗ -algebra and hence isisometric [17]. Therefore, the full, reduced, and fundamental C ∗ -algebras of G s ( X, ϕ, Q ) areall isomorphic and S ( X, ϕ, Q ) is simple. For an irreducible Smale space, it can be shown thatthere is a canonical decomposition of X into a finite number of distinct mixing componentsthat are cyclically permuted by ϕ so that X s ( P ) and X u ( Q ) are dense in each component,this remarkable fact is proven in both [15] and [20]. Therefore, π is faithful and S ( X, d, ϕ ) is
PECTRAL TRIPLES FOR HYPERBOLIC DYNAMICAL SYSTEMS 7 a direct sum of a finite number of simple components. We note that S ( X, ϕ, Q ) is separable,nuclear, and stable [16, 14].Each element of f ∈ C c ( G s ( X, ϕ, Q )) can be written as a finite sum of functions withsupport in a neighbourhood base set of the form V s ( v, w, h s , δ ). We use functions of thisform so often in the sequel that we completely describe them in the following lemma, whichfollows from the definitions. Lemma 3.5.
Suppose a is a function in C c ( G s ( X, ϕ, Q )) with support on the basic set V s ( v, w, h s , δ ) with v ∼ s w , v, w ∈ X u ( Q ) and h s : X u ( w, δ ) → X u ( v, δ ′ ) a local homeo-morphism. Then, for δ x ∈ H , π ( a ) δ x = (cid:26) a ( h s ( x ) , x ) δ h s ( x ) if x ∈ X u ( w, δ ) and h s ( x ) ∈ X u ( v, δ ′ )0 if x / ∈ X u ( w, δ ) . Define
Source ( a ) ⊆ X u ( w, δ ) to be the points for which a is non-zero on its domain. We note that every element in S ( X, ϕ, Q ) can be uniformly approximated by a finite sumof functions supported in a neighbourhood base set. We will usually begin by proving resultsusing these functions and then appealing to continuity for the general result.3.3.
The Stable Ruelle Algebra of a Smale Space.
A brief construction of the stableRuelle algebra is given. The Ruelle algebras were first constructed in [21] and alternativeconstructions were given in [14] and [16] along with many remarkable properties of these C ∗ -algebras. We also note that the stable and unstable Ruelle algebras were shown to satisfy anoncommutative version of Spanier-Whitehead duality in [9].Given an irreducible Smale space ( X, d, ϕ ), the homeomorphism ϕ : X → X induces anautomorphism α on the C ∗ -algebra S ( X, ϕ, Q ) by α ( a )( x, y ) = a ( ϕ − ( x ) , ϕ − ( y ))where a is in S ( X, ϕ, Q ) and ( x, y ) are in G s ( X, ϕ, Q ). The homeomorphism ϕ also inducesa canonical unitary on the Hilbert space H = ℓ ( X h ( P, Q )) via uδ x = δ ϕ ( x ) . Routine calculations show that ( π, u ) are a covariant representation for (
S, α ). Definition 3.6 ([14]) . The stable Ruelle algebra is the crossed product S ( X, ϕ, Q ) ⋊ α Z . Occasionally, we supress the dependence on Q and write S ⋊ α Z .4. Spectral Triples on Smale spaces
Spectral Triples.
Here we define a spectral triple and state some general propertiesof spectral triples used in the sequel. For a general reference to spectral triples see [4].To simplify notation we begin to employ [ a, b ] to denote the commutator ab − ba . Definition 4.1. A spectral triple ( A, H , D ) consists of (i): a separable Hilbert space H , (ii): a ∗ -algebra A of bounded operators on H , (iii): an unbounded self-adjoint operator D on H such that: MICHAEL F. WHITTAKER (a): the set { a ∈ A | [ D, a ] ∈ B ( H ) } is norm dense in A and (b): the operator a (1 + D ) − is a compact operator on H for all a in A .We note that the condition a (1 + D ) − is a compact operator on H , for all a in A , canbe replaced with (1 + D ) − is a compact operator on H , when A is unital. Definition 4.2.
Suppose ( A, H , D ) is a spectral triple over a unital C ∗ -algebra A withTr((1 + D ) − p ) < ∞ for some positive number p . Then the spectral triple is said to be p -summable . Furthermore,the value dim S (( A, H , D )) := inf { p > | Tr((1 + D ) − p ) < ∞} is called the spectral dimension of the spectral triple. We call ( A, H , D ) θ -summable if, forall t >
0, Tr( e − t (1+ D ) ) < ∞ . For spectral triples coming from non-unital C ∗ -algebras the definitions of summability aremuch more complex. See [18] for details. However, in the case we are interested in, wherethe C ∗ -algebra is S ( X, ϕ, Q ), the definition simplifies (since S ( X, ϕ, Q ) has local units and X u ( Q ) is the unit space of the groupoid). For S ( X, ϕ, Q ), the spectral triple ( S, H , D ) is p -summable if, for all a in C c ( X u ( Q )),Tr( a (1 + D ) − p ) < ∞ and it is θ -summable if, for all a in C c ( X u ( Q )) and for all t > ae − t (1+ D ) ) < ∞ . Spectral Triples for Smale Spaces.
We wish to construct spectral triples on thestable C ∗ -algebras of a Smale space which are geometric and encode the dynamics in anatural way. We begin by constructing a function on X s ( P ) and use this function to definea spectral triple on S ( X, ϕ, Q ).Let P and Q be finite, mutually distinct, ϕ -invariant sets of periodic points. From thispoint forward, assume that S ( X, ϕ, Q ) is represented on H = ℓ ( X h ( P, Q )) in order tosimplify notation.Select 0 < ε ≤ λ − ε X / λ > X, d, ϕ ). We aim to define a function ω : X s ( P ) → [0 , X s ( P, ε )and X s ( P ) \ ϕ − ( X s ( P, ε )) and observe that these two sets are disjoint. Now an applicationof Urysohn’s lemma implies that there exists a continuous function ω : X s ( P ) → [0 , ω ( x ) = 0 for all x ∈ X s ( P, ε ), and ω ( x ) = 1 for all x ∈ X s ( P ) \ ϕ − ( X s ( P, ε )).We remark that in practice we may define ω as desired on the complement of our two closedsets, but at this point we merely require that a continuous function exists. A typical function ω is illustrated in Figure 3, where the notation appearing in the figure is defined as follows. PECTRAL TRIPLES FOR HYPERBOLIC DYNAMICAL SYSTEMS 9 X s ( p ) pω E − E E E E E Ω p Ω cp Figure 3.
The function ω p for some p in P Notation.
We define the following sets that anticipate the constructions in the sequel andare natural in that context. Ω P = X s ( P, ε ) \ P, Ω cP = X s ( P ) \ X s ( P, ε ) ,E = ϕ − ( X s ( P, ε )) ∩ Ω cP ,E N = ϕ − N ( E ) . Let us make some remarks about these sets. Observe that ω ( x ) = 0 for x ∈ Ω P and ω ( x ) ≥ x ∈ Ω cP . In particular, the function ω is defined to either 0 or 1 on the pointsof closure of E . Also note that [ N ∈ Z E N = X s ( P ) \ P. Using ω allows us to encode the dynamics in a natural manner. Let x be a point in X s ( P ) \ P . We aim to define a function which essentially counts the number of iterationsit requires for x to be drawn into Ω P if it begins in Ω cP and subtracts to number of inverseiterations it requires for x to be removed from Ω P if it begins in Ω P . To that end, define ω s : X s ( P ) \ P → R via ω s ( x ) = ∞ X n =0 ω ◦ ϕ n ( x ) − ∞ X n =1 (1 − ω ) ◦ ϕ − n ( x ) . The function ω s , arising from the function ω in figure 3, is illustrated in Figure 4. Thefollowing Lemma summarizes the essential properties of ω s . Lemma 4.3.
Suppose P is a finite, ϕ -invariant set of periodic points in a Smale space ( X, d, ϕ ) and ω s : X s ( P ) \ P → R is defined as above. Then, (1) ω s ( x ) ≤ for x ∈ Ω P and ω s ( x ) ≥ for x ∈ Ω cP , (2) ω s ◦ ϕ − ω s = 1 , (3) ω s ( x ) = ω ◦ ϕ N ( x ) + N for x ∈ E N , and (4) ω s is continuous on X s ( P ) \ P .Proof. First, suppose x ∈ E − N for some N ∈ N . Then, the sum on the left, in the definitionof ω s , is zero since ω ( x ) = 0 and ϕ ( x ) ∈ E − N − . The sum on the right is finite since p X s ( p ) ω s E − E E E E E Figure 4.
The function ω s for some p in Pϕ N ( x ) ∈ E and 1 − ω ( ϕ n ( x )) = 0 for all n ≥ N + 1. Moreover, we have the calculation ω s ( x ) = − ∞ X n =1 (1 − ω ) ◦ ϕ − n ( x ) = − ( N − − (1 − ω )( ϕ N ( x )) = ω ( ϕ N ( x )) − N. On the other hand, suppose x ∈ E N for some N ∈ N ∪
0. Then, the sum on the right, in thedefinition of ω s , is zero since 1 − ω ( x ) = 0 and ϕ − ( x ) ∈ E N +1 . The sum on the left is finitesince ϕ N ( x ) ∈ E and ω ( ϕ n ( x )) = 0 for all n ≥ N + 1. Moreover, we have the calculation ω s ( x ) = ∞ X n =0 ω ◦ ϕ n ( x ) = ω ( ϕ N ( x )) + N. This proves that ω s is well-defined and the first three statements in the Lemma. For thefourth, we observe that ω s ( x ) is continuous on E N , for all N ∈ Z , since ω is continuous on E . If E k ∩ E k +1 = r , for k ∈ Z , it follows, from the definition of ω s , that ω s ( r ) = k + 1.Since S N ∈ Z E N = X s ( P ) \ P and ω s gives equal value to the common boundary of E k and E k +1 for all k ∈ N , ω s is continuous. (cid:3) We now consider how the function ω s interacts with functions in C c ( G s ( X, ϕ, Q )) supportedon basic sets, see Lemma 3.5 for details on basic sets and for a definition of
Source ( a ) for a in S ( X, ϕ, Q ). Lemma 4.4.
Suppose P is a finite, ϕ -invariant set of periodic points in a Smale space ( X, d, ϕ ) and ω s : X s ( P ) → R is defined as above. Let a ∈ C c ( G s ( X, ϕ, Q )) be supportedon a basic set V s ( v, w, h s , δ ) so that Source ( a ) ⊆ X u ( w, δ ) and h s ( Source ( a )) ⊆ X u ( v, δ ′ ) .Then, PECTRAL TRIPLES FOR HYPERBOLIC DYNAMICAL SYSTEMS 11 (1) there exists N ∈ Z such that E n ∩ Source ( a ) = ∅ for all n ≤ N , (2) for all N , the number of points in E N ∩ Source ( a ) is finite, (3) if x ∈ E N ∩ Source ( a ) , then there exists K ∈ N such that h s ( x ) ∈ S Kk = − K E N + k .Proof. Since
Source ( a ) is a pre-compact subset of X u ( Q ) and P * X u ( Q ), define δ > δ = inf { d ( p, x ) | p ∈ P, x ∈ Source ( a ) } . Now there exists N ∈ Z such that E n ⊂ X s ( P, δ )for all n ≤ N . Therefore, E n ∩ Source ( a ) = ∅ for all n ≤ N as well. For the second claim, Source ( a ) and E N are transverse and compact for all N , it follows that E N ∩ Source ( a ) isfinite.For the third claim, first note that given that a is supported in V s ( v, w, h s , δ ), there exists M such that for all x ∈ Source ( a ), d ( ϕ M ( h s ( x )) , ϕ M ( x )) < ε X / ϕ M ( x ) , ϕ M ( h s ( x ))] = ϕ M ( x ). Moreover, it follows that there exists L ∈ N such that D > ε X where D =sup { d ( y, y ′ ) | y, y ′ ∈ E L and [ y, y ′ ] = y } ; that is, we can find E L so that E L has diam-eter larger than ε X on the stable set of each periodic point p in P . Now we claim thatif ( h s ( x ) , x ) ∈ V s ( v, w, h s , δ ) has the property that x ∈ E m where m ≥ M + L + 1, then h s ( x ) ∈ S k = − E m + k . Indeed, for all ( h s ( x ) , x ) we have d ( ϕ M ( h s ( x )) , ϕ M ( x )) < ε X / ϕ M ( x ) is in E L +1 , then by the triangle inequality we have ϕ M ( h s ( x )) ∈ ∪ k = − E L +1+ k . Nowapplying ϕ − M to ϕ M ( x ) and ϕ M ( h s ( x )) proves the claim. We are left with the case that Source ( a ) ∈ E n for n < M + L + 1. However, combining part (1) and (2) implies that thereare only a finite number of such elements, so that we may define K = max { , | i − j | | h s ( x ) ∈ E i , x ∈ E j , and i, j < M + L + 1 } , which is finite. Now K has the property that if x ∈ E N ∩ Source ( a ), then h s ( x ) ∈ S Kk = − K E N + k . (cid:3) A θ -Summable Spectral Triple. In this section, we define a spectral triple on S ( X, ϕ, Q ). The idea is to use ω s to define a Dirac operator on H = ℓ ( X h ( P, Q )). Let Dδ x = ω s ( x ) δ x . The domain of D is given by Domain ( D ) = { ξ | X x ∈ X h ( P,Q ) ω s ( x ) | ξ ( x ) | < ∞} . and routine calculations show that D is self-adjoint and unbounded. Lemma 4.5.
For a ∈ C c ( G s ( X, ϕ, Q )) , the commutator [ D, a ] is a bounded operator on H .Proof. Let a in C c ( G s ( X, ϕ, Q )) be supported on a basic set V s ( v, w, h s , δ ). By part (3) ofLemma 4.4, if x ∈ E N ∩ Source ( a ), then there exists K ∈ N such that h s ( x ) ∈ S Kk = − K E N + k .Therefore, for any x ∈ E N ∩ Source ( a ), using part (3) of lemma 4.3, we compute k [ D, a ] δ x k = k ( ω s ( h s ( x )) − ω s ( x )) a ( h s ( x ) , x ) δ h s ( x ) k = | ω s ( h s ( x )) − ω s ( x ) || a ( h s ( x ) , x ) |≤ | ω ( ϕ N + K ( h s ( x ))) + N + K − ( ω ( ϕ N ( x )) + N ) || a ( h s ( x ) , x ) |≤ ( K + 1) | a ( h s ( x ) , x ) | . Since a is compactly supported, | a ( h s ( x ) , x ) | attains a maximum value. Moreover, everythingabove is independent of N so that [ D, a ] is bounded. For the general case we recall that anyelement of C c ( G s ( X, ϕ, Q )) is in the span of functions supported on basic sets. (cid:3)
Proposition 4.6.
For every a in S ( X, ϕ, Q ) , the operator a (1 + D ) − is compact on H .Proof. Let a in S ( X, ϕ, Q ) be supported on a basic set of the form V s ( v, w, h s , δ ). Bypart (1) of Lemma 4.4, there exists M such that E m ∩ Source ( a ) = ∅ for all m ≤ M .Furthermore, by part (2) of Lemma 4.4, the number of elements in E N ∩ Source ( a ) is finitefor all N . Now using part (3) of Lemma 4.3, for x ∈ E N ∩ Source ( a ) we have k a (1 + D ) − δ x k = k a ( h s ( x ) , x )1 + ω s ( x ) δ sh ( x ) k ≤ | a ( h s ( x ) , x ) | N . Since a has compact support, let A = sup {| a ( h s ( x ) , x ) | | x ∈ Source ( a ) } . Moreoversince h s takes basis vectors to basis vectors and is a homeomorphism from Source ( a ) to h s ( Source ( a )) it follows that, restricted to E N , k a (1 + D ) − k ≤ A N . Therefore, a (1 + D ) − is a norm limit of finite rank operators. Moreover, a in S ( X, ϕ, Q ) isa norm limit of finite sums of operators of the form a , so a (1 + D ) − is compact as well. (cid:3) Before arriving at our main theorem for the section, we must delve into a technical result.For an irreducible Smale space (
X, d, ϕ ), the topological entropy of (
X, d, ϕ ) is denoted h ( X, ϕ ) and is the growth rate of the number of essentially different orbit segments of length N , for further details see [2]. We state the following result which is obtained by combiningLemma 5 . .
12 in [10]. There are also several similar results in [11].
Theorem 4.7 ([10]) . Suppose ( X, d, ϕ ) is an irreducible Smale space with P and Q distinct,finite, ϕ -invariant sets of periodic points. Then for any δ , δ > and any w ∈ X u ( Q ) , { x | ϕ − N ( X s ( P, δ )) ∩ X u ( w, δ ) } is finite. Moreover, lim N →∞ | N log( { x | ϕ − N ( X s ( P, δ )) ∩ X u ( w, δ ) } ) − h ( X, ϕ ) | = 0 . Theorem 4.8.
Suppose ( X, d, ϕ ) is an irreducible Smale space, then ( S, H , D ) is a non-unital, θ -summable spectral triple.Proof. We have shown that ( S, H , D ) is a spectral triple. It remains to show that ( S, H , D )is θ -summable. We must show that, for a in C c ( X u ( Q )) a positive operator, we haveTr( ae − t (1+ D ) ) < ∞ for all t >
0. By part (1) of Lemma 4.4, there exists M such that E m ∩ Source ( a ) = ∅ for all m ≤ M . Furthermore, by part (2) of Lemma 4.4, the numberof elements in E N ∩ Source ( a ) is finite for all N . Now using part (3) of Lemma 4.3, for PECTRAL TRIPLES FOR HYPERBOLIC DYNAMICAL SYSTEMS 13 x ∈ E N ∩ Source ( a ) we haveTr( ae − t (1+ D ) ) = X x ∈ X h ( P,Q ) D ae − t (1+ D ) δ x , δ x E = X x ∈ Source ( a ) a ( x, x ) e t (1+ ω s ( x )) ≤ ∞ X n = M X { x | x ∈ E n ∩ Source ( a ) } a ( x, x ) e t (1+( n ) ) . (1)Now from Theorem 4.7, for ε >
0, there exists N such that for all n ≥ N , { x | x ∈ E n ∩ Source ( a ) } < e n ( h ( X,ϕ )+ ε ) . Therefore, letting A = sup { a ( x, x ) | x ∈ Source ( a ) } , we have X { x | x ∈ E n ∩ Source ( a ) } a ( x, x ) e t (1+( n ) ) < Ae n ( h ( X,ϕ )+ ε ) e t (1+( n ) ) = Ae n ( h ( X,ϕ )+ ε ) − t (1+( n ) ) < Ae n ( h ( X,ϕ )+ ε ) − tn = Ae n ( h ( X,ϕ )+ ε − tn ) . Putting this into (1) and letting R denote the first N − ae − t (1+ D ) ) = R + ∞ X n = M Ae n ( h ( X,ϕ )+ ε − tn ) , which converges since we can choose N sufficiently large that tN > h ( X, ϕ ) + ε . (cid:3) Recall that the stable Ruelle algebra is the crossed product S ⋊ α Z , see Section 3.3.As operators, S ⋊ α Z is the completion of span { a · u k | a ∈ S ( X, ϕ, Q ) and k ∈ Z } inthe Hilbert space H = ℓ ( X h ( P, Q )) where u is the canonical unitary on H defined by uδ x = δ ϕ ( x ) . Using (2) in Lemma 4.3, we have [ u, D ] = u so that k [ u, D ] k = 1. Therefore,[ a · u k , D ] = a [ u k , D ] + [ a, D ] u k is a bounded operator and we obtain a spectral triple on thestable Ruelle algebra as well. Theorem 4.9.
Suppose ( X, d, ϕ ) is an irreducible Smale space, then ( S ⋊ α Z , H , D ) is anon-unital, θ -summable spectral triple. A p -Summable Spectral Triple. In this section we add the hypothesis that thefunction ω is locally Lipschitz continuous in order to define a summable spectral triple on S ( X, ϕ, Q ). The added assumption that ω is locally Lipschitz continuous will not restrictthe Smale spaces we consider in any way since a locally Lipschitz continuous function canbe defined using the Smale space metric.Let us define ω to be locally Lipschitz continuous; that is, there exists a constant C suchthat if x, y ∈ E , [ x, y ] = x , and d ( x, y ) < ε X /
2, then | ω ( x ) − ω ( y ) | < C d ( x, y ) where the metric comes from the Smale space itself. In fact, since ( X, d ) is a compact metricspace we can always define such a function using the metric and regarding E as a disjointunion of closed sets, one for each element of P . Let us also define a constant C s = 2 KC where K = max { k > | [ x, y ] = x, d ( x, y ) < ε X / , with x ∈ E and y ∈ E k } . Lemma 4.10.
The function ω s is locally Lipschitz continuous on ∪ ∞ n =0 E n ; that is, if x, y ∈∪ ∞ n =0 E n , d ( x, y ) < ε X / and [ x, y ] = x , then | ω s ( x ) − ω s ( y ) | < C s d ( x, y ) . Proof.
First observe that ω s ( x ) is locally Lipschitz continuous, with Lipschitz constant C ,on E N , for all N ∈ N . Indeed, suppose x, y ∈ E N such that [ x, y ] = x . Then, using part (3)of Lemma 4.3, | ω s ( x ) − ω s ( y ) | = | ( ω ( ϕ N ( x )) + N ) − ( ω ( ϕ N ( y )) + N ) | = | ω ( ϕ N ( x )) − ω ( ϕ N ( y )) | < C d ( ϕ N ( x ) , ϕ N ( y )) < C λ − N d ( x, y ) . Now suppose x, y ∈ ∪ ∞ n =0 E n , d ( x, y ) < ε X / x, y ] = x . Then we note that if x ∈ E N then y ∈ ∪ K + Nk = − K + N E k where K comes from the definition of C s . The triangle inequality givesthe desired result. (cid:3) Define an operator D on H = ℓ ( P, Q ) via D δ x = λ ω s ( x ) δ x where λ > X, d, ϕ ). The operator D is also self-adjoint, un-bounded, and has dense domain. Lemma 4.11.
For a in C c ( G s ( X, ϕ, Q )) , the commutator [ a, D ] is a bounded operator on H .Proof. Let a in C c ( G s ( X, ϕ, Q )) be supported on a basic set V s ( v, w, h s , δ ), which implies thatthere exists M such that, for all ( h s ( x ) , x ) ∈ V s ( v, w, h s , δ ), we have d ( ϕ M ( h s ( x )) , ϕ M ( x )) <ε X /
2. By Lemma 4.4, for all but a finite number of ( h s ( x ) , x ) ∈ V s ( v, w, h s , δ ) we have both x and h s ( x ) in ∪ ∞ m = M E m . Suppose we are given such an ( h s ( x ) , x ). Without loss of generality PECTRAL TRIPLES FOR HYPERBOLIC DYNAMICAL SYSTEMS 15 suppose x ∈ E k and h s ( x ) ∈ E j where M ≤ k ≤ j . We compute k [ a, D ] δ x k = k ( λ ω s ( x ) − λ ω s ( h s ( x )) ) a ( h s ( x ) , x ) δ h s ( x ) k = | λ ω s ( x ) − λ ω s ( h s ( x )) || a ( h s ( x ) , x ) | = λ ω s ( x ) | − λ ω s ( h s ( x )) − ω s ( x ) || a ( h s ( x ) , x ) | = λ k + ω ( ϕ k ( x )) | − λ ω s ( M + ϕ M ( h s ( x ))) − ( M + ω s ( ϕ M ( x ))) || a ( h s ( x ) , x ) |≤ λ k +1 | − λ ω s ( ϕ M ( h s ( x ))) − ω s ( ϕ M ( x )) || a ( h s ( x ) , x ) | < λ k +1 | − λ C s d ( ϕ M ( h s ( x )) ,ϕ M ( x )) || a ( h s ( x ) , x ) | < λ k +1 | − λ C s ε X / || a ( h s ( x ) , x ) | < λ k +1 | − λ log λ (1+ C s ε X / || a ( h s ( x ) , x ) | = λ k +1 | − (1 + C s ε X / || a ( h s ( x ) , x ) | = λ k +1 C s ε X / | a ( h s ( x ) , x ) |≤ λ k +1 λ M − k C s ε X / | a ( h s ( x ) , x ) | = C s λ M +1 ε X / | a ( h s ( x ) , x ) | where M depend only on the set V s ( v, w, h s , δ ). Since a is compactly supported it attainsits maximum. Thus, in this case [ a, D ] is bounded.On the other hand, if ( h s ( x ) , x ) is in the finite set where both x and h s ( x ) are not in ∪ ∞ m = M E m , then we can take the maximum value of | λ ω s ( x ) − λ ω s ( h s ( x )) | which is boundedsimply because it is a finite set. Therefore, [ a, D ] is bounded and the Lemma is proven. (cid:3) To complete the proof that ( S, H , D ) is a non-unital spectral triple we need only showthat a (1 + D ) − is a compact operator for every a in S ( X, ϕ, Q ). The same argumentas presented in Section 4.3 gives the result. We will now show that ( S, H , D ) is a finitelysummable spectral triple. Indeed, ( S, H , D ) is log λ ( e ) h ( X, ϕ )-summable, where h ( X, ϕ ) isthe topological entropy of the Smale space (
X, d, ϕ ). We note that the factor log λ ( e ) ismerely a base change from a base e logarithm to a base λ logarithm. Theorem 4.12.
Suppose ( X, d, ϕ ) is an irreducible Smale space, then ( S, H , D ) is a non-unital, log λ ( e ) h ( X, ϕ ) -summable spectral triple, where h ( X, ϕ ) is the topological entropy ofthe Smale space ( X, d, ϕ ) .Proof. We have shown that ( S, H , D ) is a spectral triple. It remains to show that ( S, H , D )is summable. We must show that, for a in C c ( X u ( Q )) a positive operator, we haveTr( a (1 + D ) − s ) < ∞ for some s >
0. By part (1) of Lemma 4.4, there exists M such that E m ∩ Source ( a ) = ∅ for all m ≤ M . Furthermore, by part (2) of Lemma 4.4, the number of elements in E N ∩ Source ( a ) is finite for all N . Now using part (3) of Lemma 4.3, for x ∈ E N ∩ Source ( a ) we haveTr( a (1 + D ) − s ) = X x ∈ X h ( P,Q ) (cid:10) a (1 + D ) − s ) δ x , δ x (cid:11) = X x ∈ Source ( a ) a ( x, x )(1 + λ ω s ( x ) ) s/ ≤ ∞ X n = M X { x | x ∈ E n ∩ Source ( a ) } a ( x, x )(1 + λ n ) s/ . (2)Now from Theorem 4.7, for ε >
0, there exists N such that for all n ≥ N , e n ( h ( X,ϕ ) − ε ) < { x | x ∈ E n ∩ Source ( a ) } < e n ( h ( X,ϕ )+ ε ) . (3)Therefore, letting A = sup { a ( x, x ) | x ∈ Source ( a ) } , we have X { x | x ∈ E n ∩ Source ( a ) } a ( x, x )(1 + λ n ) s/ < Ae n ( h ( X,ϕ )+ ε ) (1 + λ n ) s/ < Ae n ( h ( X,ϕ )+ ε ) ( λ n ) s/ = Aλ log λ ( e ) n ( h ( X,ϕ )+ ε ) ( λ sn )= Aλ n (log λ ( e ) h ( X,ϕ )+log λ ( e ) ε ) − sn = Aλ n ((log λ ( e ) h ( X,ϕ )+log λ ( e ) ε ) − s ) Putting this into (2) and letting R denote the first N − a (1 + D ) − s ) < R + ∞ X n = M Aλ n ((log λ ( e ) h ( X,ϕ )+log λ ( e ) ε ) − s ) , which converges geometrically for s > log λ ( e ) h ( X, ϕ ) + log λ ( e ) ε . Since this holds for any ε > λ ( e ) h ( X, ϕ ) ≥ inf { s | Tr( a (1 + D ) − s ) < ∞} . On the other hand, using the other inequality in (3), a similar computation shows thatTr( a (1 + D ) − s ) > R + ∞ X n = M min { a ( x, x ) } λ s λ n ((log λ ( e ) h ( X,ϕ ) − log λ ( e ) ε ) − s ) , which converges geometrically only if s > log λ ( e ) h ( X, ϕ ) − log λ ( e ) ε for every ε >
0. There-fore, we have log λ ( e ) h ( X, ϕ ) = inf { s | Tr( a (1 + D ) − s ) < ∞} . (cid:3) To conclude, we note that it is not obvious that the operator D gives rise to a spectraltriple on the stable Ruelle algebra S ⋊ α Z . We would be very interested to know if it does. References [1] S. Baaj and P. Julg,
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MICHAEL F. WHITTAKER, School of Mathematics, University of Wollongong, Aus-tralia
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