Accessibility and centralizers for partially hyperbolic flows
AACCESSIBILITY AND CENTRALIZERS FOR PARTIALLY HYPERBOLIC FLOWS
TODD FISHER AND BORIS HASSELBLATTA
BSTRACT . Stable accessibility for partially hyperbolic diffeomorphisms is central totheir ergodic theory, and we establish its C -density among • all, • volume-preserving,• symplectic, and • contact partially hyperbolic flows .As applications, we obtain in each of these 4 categories C -density of C -stable topological transitivity, ergodicity, and triviality of the centralizer.
1. I
NTRODUCTION
The renaissance of partial hyperbolicity that began in the 1990s centered on the questfor stably ergodic dynamical systems. The Hopf argument as the central technical devicebrought the notion of accessibility to the fore, and this motivated results to the effect thatstable accessibility of partially hyperbolic diffeomorphisms (Definition 2.1) is C -dense[20, Main Theorem], [5], [29, Theorem 8.5]. Our first aim is to show that this also holdsfor flows. Among the applications is a C -open dense set of partially hyperbolic flowswith trivial flow-centralizer. Statement of results.Theorem 1.1 (Generic accessibility) . For any smooth compact manifold M and r ≥ ,C -stable accessibility (Definition 2.3) is C dense among • all, • volume-preserving, • symplectic, and • contact partially hyperbolic C r flows on M. Our motivation for establishing genericity of accessibility was to adapt arguments ofBurslem [16] in order to establish generic triviality of the flow-centralizer (Definition 1.5),in particular a relative paucity of faithful (cid:82) k Anosov actions.
Theorem 1.2 (No centralizer) . On any smooth compact manifold M and for any r ≥ , C r -flows which C -stably have trivial flow-centralizer are C dense among • all, • volume-preserving, • symplectic, and • contact partially hyperbolic C r flows on M. There are more direct applications of accessibility:
Corollary 1.3 (Generic transitivity) . On any smooth compact manifold M and for anyr ≥ , the set of topologically transitive C r -flows is C open and dense among • vol-ume-preserving • symplectic • contact partially hyperbolic C r flows on M.Proof. Theorem 1.1 provides an open dense set of accessible such flows; their time-1maps are accessible volume-preserving partially hyperbolic diffeomorphisms for whichalmost every point has a dense orbit [12], [29, Theorem 8.3]; those points then have denseflow-orbits. (cid:3)
Mathematics Subject Classification.
Key words and phrases.
Dynamical systems; partially hyperbolic systems and dominated splittings; central-izer; accessibility.T.F. is supported by Simons Foundation grant a r X i v : . [ m a t h . D S ] J u l TODD FISHER AND BORIS HASSELBLATT
Indeed, strong ergodic properties, such as the K-property [21, Definition 3.4.2], aresimilarly common if one adds the assumption of center-bunching [15]:
Corollary 1.4 (K-property) . On any smooth compact manifold M and for any r ≥ , theset of C r -flows for which the natural volume has the K-property (for all time-t maps) is C open and dense among • volume-preserving • symplectic • contact center-bunchedpartially hyperbolic C r flows on M.Proof. Theorem 1.1 provides an open dense set of accessible such flows; for any t (cid:54)= t maps are accessible volume-preserving center-bunched partially hyperbolicdiffeomorphisms and hence have the K-property [15, Theorem 0.1]. (cid:3) As these corollaries indicate, accessibility is central to the ergodic theory of partiallyhyperbolic dynamical systems (Definition 2.1), the theory of which was revived 2 decadesafter its founding [13,14] when Pugh and Shub sought nonhyperbolic examples of volume-preserving dynamical systems that are stably ergodic [23, 30–32]. Stable ergodicity wasestablished by adapting the Hopf argument [21, §7.1,2], and accessibility is the keyingredient [15].In this context, flows (as opposed to diffeomorphisms) have received little attention:accessibility of a partially hyperbolic flow and of its time-1 map are equivalent, so a theoryfor diffeomorphisms suffices for establishing ergodicity for partially hyperbolic flows;indeed, the initial examples of stably ergodic partially hyperbolic diffeomorphisms weretime-1 maps of hyperbolic flows. Moreover, the stability of ergodicity and accessibility ofthe time-1 map implies their stability for the flow.Considering flows becomes salient, however, when investigating the prevalence of(stable) accessibility and ergodicity. Diffeomorphisms are rarely the time-1 map of aflow, so density or genericity results for diffeomorphisms do not automatically imply likeresults for flows. The issue is that one needs to argue that the flows themselves ratherthan just their time-1 maps can be perturbed in a desired fashion.
Centralizers.
Our interest in accessibility arose from a desire to understand the cen-tralizer of flows beyond hyperbolic ones [6, 21]. Partially hyperbolic flows are a naturalnext step, and this led us to wanting to adapt the pertinent result [16, Theorem 1.2] toflows—and its proof uses accessibility in an essential way.The centralizer of a dynamical system reflects the symmetries of that system, and thisleads to the expectation that the centralizer of a (sufficiently complex, see [21, Example1.8.8]) dynamical system is often small. (It also reflects nonuniqueness of conjugacies[21, p. 97].) Since the notion is relative to an ambient group [21, Remark 1.8.9], we makethe needed terminology explicit.
Definition 1.5.
The flow-centralizer of a C r flow consists of the C r flows that commutewith it, and there are different types of triviality [26]. We say that the centralizer is trivial ifit consists of constant scalings of the flow, i.e., the generating vector fields of commutingflows are constant scalings of the given vector field; in this case we also say that the flowis self-centered .A flow has quasi-trivial flow-centralizer if commuting vector fields are related to thegiven one by a smooth scalar factor. Remark 1.6 (Hyperbolicity implies small centralizer [21, §9.1]) . An Anosov flow hastrivial flow-centralizer [21, Corollary 9.1.4], and this extends to kinematic-expansiveflows on a connected space with at most countably many chain-components, all ofwhich are topologically transitive [21, Theorem 9.1.3].
CCESSIBILITY AND CENTRALIZERS FOR FLOWS 3
Quasi -triviality of the centralizer holds for (Bowen-Walters) expansive flows [28] andindeed C r -generic flows [26] (including volume-preserving ones [11]).An open and dense subset of C ∞ Axiom-A flows with a strong transversality conditionhas (properly) trivial flow-centralizer [33], as do transitive Komuro expansive flows [9](this includes the Lorenz attractor), and C -generic sectional Axiom-A flows [8, 10].Indeed, hyperbolic flows usually have small centralizers [21, Theorem 9.1.3], and weextend this and the requisite accessibility result to partially hyperbolic flows.One can also look for diffeomorphisms that commute with a flow; the set of theseis the diffeomorphism-centralizer of the flow. Even Anosov flows can have nontrivialdiffeomorphism-centralizers [27, Section 5].While there has been interesting work beyond the hyperbolic context (the centralizeris quasi-trivial for a C -generic flow with at most finitely many sinks or sources [27],trivial if the flow moreover has at most countably many chain-recurrence classes), ourresults produce open dense sets with the desired properties, whereas elsewhere, oftenonly “residual” is known.If one thinks of the centralizer question as the possibility of embedding a flow in afaithful (cid:82) -action (or a diffeomorphism into a faithful (cid:90) -action), then a deeper probecould focus on the classification (or rigidity) of (cid:82) k -actions for k ≥
2, e.g., aiming toshow that they are necessarily algebraic. Great efforts have already been devoted to thisaim [19, 24], and quite recently, these have been pushed into the partially hyperbolicrealm—for discrete time (for smooth, ergodic perturbations of certain algebraic systems,the smooth centralizer is either virtually (cid:90) l or contains a smooth flow [7, 18]). Also,the centralizer of a partially hyperbolic (cid:84) -diffeomorphism homotopic to an Anosovautomorphism is virtually trivial unless the diffeomorphism is smoothly conjugate to itslinear part [22]. Acknowledgements.
We would like to thank Sylvain Crovisier, Jana Rodríguez Hertz,Raul Ures, and Amie Wilkinson for helpful suggestions.2. B
ACKGROUND
In this section we review needed definitions and previous results. [21] provides muchof the basic background, but we define two main notions here.
Definition 2.1 (Partially hyperbolic) . An embedding f is said to be (strongly) partiallyhyperbolic on a compact f -invariant set Λ if there exist numbers C > < λ ≤ µ < λ ≤ µ < λ ≤ µ with µ < < λ and an invariant splitting into nontrivial stable, central and unstable subbundles(2) T x M = E s ( x ) ⊕ E c ( x ) ⊕ E u ( x ), d x f E τ ( x ) = E τ ( f ( x )), τ = s , c , u such that if n ∈ (cid:78) , then C − λ n ≤ (cid:107)(cid:98) d x f n (cid:22) E s ( x ) (cid:99)(cid:107) ≤ (cid:107) d x f n (cid:22) E s ( x ) (cid:107) ≤ C µ n , C − λ n ≤ (cid:107)(cid:98) d x f n (cid:22) E c ( x ) (cid:99)(cid:107) ≤ (cid:107) d x f n (cid:22) E c ( x ) (cid:107) ≤ C µ n , C − λ n ≤ (cid:107)(cid:98) d x f n (cid:22) E u ( x ) (cid:99)(cid:107) ≤ (cid:107) d x f n (cid:22) E u ( x ) (cid:107) ≤ C µ n .In this case we set E cs : = E c ⊕ E s and E cu : = E c ⊕ E u .A flow is said to be partially hyperbolic on a compact flow-invariant set if its time-1map is partially hyperbolic on it, and uniformly hyperbolic if the center direction of the TODD FISHER AND BORIS HASSELBLATT time-1 map consists only of the flow direction. In either case we say that a dynamicalsystem is partially hyperbolic on an invariant set.For a partially hyperbolic set Λ and x ∈ Λ there exist local stable and unstable man-ifolds that define global stable and unstable manifolds denoted by W s ( x ) and W u ( x )respectively. Remark 2.2 (Persistence of partial hyperbolicity) . For a flow Φ and a partially hyper-bolic set Λ for Φ with splitting T Λ M = E u ⊕ E c ⊕ E s and continuous invariant cone fields C u , C s , C cu and C cs , containing E u , E s , E cu and E cs , respectively, there exist neighbor-hoods U of Λ and U of Φ and cone fields C u , C s , C cu , and C cs over U such thatif Ψ ∈ U and Λ (cid:48) ⊂ U is a compact Ψ -invariant set, then Λ (cid:48) is partially hyperbolicwith a splitting T ∆ M = E s Ψ ⊕ E c Ψ ⊕ E u Ψ such that E u Ψ , E s Ψ , E cu Ψ , and E cs Ψ are contained in C u , C s , C cu , and C cs , respectively.To avoid confusion we will sometimes refer to these neighborhoods as U ( Φ , Λ ) and U ( Φ , Λ ) when we consider different flows and different partially hyperbolic sets. Definition 2.3 (Accessibility) . Two points p , q in a partially hyperbolic set Λ ⊂ M are accessible if there are points z i ∈ M with z = p , z (cid:96) = q , such that z i ∈ V α ( z i − ) for i = (cid:96) and α = s or u . The collection of points z , z ,..., z (cid:96) is called the us - path connecting p and q and is denoted variously by [ p , q ] f = [ p , q ] = [ z , z ,..., z (cid:96) ]. (Notethat there is an actual path from p to q that consists of pieces of smooth curves on localstable or unstable manifolds with the z i as endpoints.)Accessibility is an equivalence relation and the collection of points accessible from agiven point p is called the accessibility class of p .A partially hyperbolic set Λ is bisaturated if W u ( x ) ⊂ Λ and W s ( x ) ⊂ Λ for all x ∈ Λ ,and a bisaturated partially hyperbolic set is said to be accessible if the accessibility classof any point is the entire set, or, in other words, if any two points are accessible.If the entire manifold is partially hyperbolic for a flow, then it is bisaturated. In thiscase, the flow is accessible if the entire manifold is an accessibility class.A pair ( Φ , Λ ) of a dynamical system and a partially hyperbolic set of it is accessible on X ⊂ M if for every p ∈ X ∩ Λ and q ∈ X there is an su -path from p to q . If Λ isbisaturated, this implies that either X ∩ Λ = ∅ or X ⊂ Λ . Furthermore, a pair ( Φ , Λ ) of adynamical system and a partially hyperbolic set of it is stably accessible on X ⊂ M if thereexist neighborhoods U of Λ and U of Φ such that if (cid:101) Φ ∈ U and (cid:101) Λ ⊂ U is a (cid:101) Φ -invariantbisaturated compact set, then ( (cid:101) Φ , (cid:101) Λ ) is accessible on X .Although we are interested in flows that are partially hyperbolic over the entire mani-fold to obtain our main results, our general result on accessibility (Theorem 3.1) holdsfor bisaturated partially hyperbolic sets.We obtain accessibility (Theorem 1.1) by adapting from [5, 20] the proof of Theorem 2.4 (Avila–Crovisier–Dolgopyat–Wilkinson [20, Main Theorem], [5, footnote p.13]) . If M is a smooth compact manifold and r ≥ , then stable accessibility is C denseamong • all • volume-preserving • symplectic partially hyperbolic C r diffeomor-phisms of M. From Theorem 1.1, we obtain Theorem 1.2 by adapting lemmas from the proof of
Theorem 2.5 (Burslem [16, Theorem 1.2]) . In the set of C r partially hyperbolic diffeomor-phisms of a compact manifold M (r ≥ ), there is a C -open and C -dense subset V whoseelements all have discrete diffeomorphism-centralizer. CCESSIBILITY AND CENTRALIZERS FOR FLOWS 5
3. A
CCESSIBILITY
The arguments of [5] apply to the time-1 map of a partially hyperbolic flow—with oneessential adaptation. We need to show that the perturbation that C -approximates apartially hyperbolic diffeomorphism by an accessible one [5, Section 2.5] can be effectedin such a way that it gives an accessible flow as an approximation of a given partiallyhyperbolic flow .Theorem 1.1 is a consequence of the following more general theorem, which corre-sponds to [5, Theorem B]. Theorem 3.1.
Let Λ be a partially hyperbolic set for a flow Φ on a closed manifold M, andlet U be a C neighborhood of Φ . There exists a neighborhood U of Λ and a nonemptyopen set O ⊂ U such that if Ψ ∈ O and ∆ ⊂ U is a bisaturated partially hyperbolic set for Ψ , then ∆ is accessible for Ψ .Furthermore, this holds among volume-preserving, symplectic, and contact flows. Definitions.
We first review notation introduced in [5] before explaining the adaptationsthat need to be made to prove Theorems 1.1 and 3.1. We will use slightly differentnotation, because we are using the notation for flows that they use for the charts.
Proposition 3.2 (Adapted charts) . Let M be a smooth manifold with dim( M ) = d. Foreach point p ∈ M there is a chart f p : B (0,1) ⊂ T p M → M with the following properties:(1) The map p (cid:55)→ f p is piecewise continuous in the C topology. So there are open setsU ,..., U (cid:96) ⊂ M and– compact sets K ,..., K (cid:96) covering M with K i ⊂ U i ,– trivializations g i : U i × (cid:82) d → T U i M such that g i ({ p } × B (0,2)) contains theunit ball in T p M for each p ∈ U i , and– smooth maps F i : U i × B (0,2) → M,such that each p ∈ M belongs to some K i , with f p = F i ◦ g − i on B (0,1) ⊂ T p M.(2) When a volume, symplectic, or contact form has been fixed on M, this pulls backunder f p to a constant (and standard such) form on T p M [25, Theorems 5.1.27,5.5.9, 5.6.6].
Remark 3.3.
We note that given a compact set K with continuous splitting T K M = E ⊕ E there is a Riemannian metric with respect to which the norm of the projection from E to E is arbitrarily small. Then the charts F can be chosen so the bundles E and E arelifted in B d (0,3) to nearly constant bundles. Remark 3.4.
For a volume, symplectic, or contact form, the standard chart expressesthat form, respectively, as • d x ∧ ··· ∧ d x d , • d /2 (cid:88) i = d x i ∧ d y i , • α = d t + ( d − (cid:88) i = x i d y i .We note that a chart of the latter type is automatically of flow-box type: the Reeb vectorfield Y of α = d t + (cid:80) ( d − i = x i d y i is ∂ / ∂ t because it is (uniquely) defined by d α ( Y , · ) ≡ α ( Y ) ≡ Λ (the exis-tence of a foliation tangent to E c ), and so we define approximate center manifolds thatwill be sufficient. TODD FISHER AND BORIS HASSELBLATT
Definition 3.5 ( c -admissible disk) . For sufficiently small η > p ∈ Λ , denote by B c (0, η ) the ball around 0 in E cp of radius η . The set V η ( p ) : = f p ( B c (0, η )) is a c-admissibledisk with radius η = : r ( V η ( p )). A c-admissible disk family is a finite collection of pairwisedisjoint, c -admissible disks.For β ∈ (0,1) let β V η ( p ) be the c -admissible disk centered at p with radius r ( β V η ( p )) = βη . Definition 3.6 (Return time) . The return time R : P ( M ) → [0, ∞ ] is defined for a set S ⊂ M that is contained in a flow-box ([21, Definition 1.1.13]) of “height” τ as the infimumof t ∈ ( τ , ∞ ] such that ϕ t ( S ) ∩ S (cid:54)= ∅ .Note that the above definition presumes that S contains no fixed point of the flow. Itimplies that if p ∈ M is not fixed, then R ( B η ( p )) −−−− η → → per( p ) if we agree that per( p ) = ∞ if p is not periodic, and per( p ) is the period of p for any periodic p .For a c -admissible disk family D and β ∈ (0,1) we let β D : = { β D : D ∈ D }, | D | : = (cid:91) D ∈ D D , r ( D ) : = sup D ∈ D r ( D ), and R ( D ) : = R ( | D | ). The Avila–Crovisier–Dolgopyat–Wilkinson arguments.
The proof of accessibility in [5]proceeds in two steps. The first is a general fact for partially hyperbolic sets for diffeo-morphisms on the existence of c -admissible disk families that stably meet all unstableand stable leaves as follows. Definition 3.7 (Global c -section) . We say that a set X ⊂ M is a (global) c-section for ( Φ , Λ )if X ∩ ∆ (cid:54)= ∅ for every bisaturated subset ∆ ⊂ Λ . Remark 3.8.
This terminology alludes to that of a (Poincaré) section for a flow, whichmeets bunches of orbits; (global) c -sections meet many stable and unstable leaves. Al-though this is not a defining property, the (global) c -sections we find will be transverse tostable and unstable leaves (Proposition 3.9) and will indeed meet all stable and unstableleaves once accessible (Proposition 3.11).Via time- t maps, the result on the existence of such families immediately holds in oursetting. Proposition 3.9 ([5, Proposition 1.4]) . Let Λ be partially hyperbolic set for Φ . Thenthere exists a δ > with the following property. If U is a neighborhood of Λ such thatU ⊂ U ( Φ , Λ ) and T > , then there exists a c-admissible disk family D and σ > suchthat:(1) r ( D ) < T − ,(2) R ( D ) > T (this implies that | D | contains no fixed point), and(3) if Ψ satisfies d ( Φ , Ψ ) < δ and d ( Φ , Ψ ) < σ , then for any bisaturated partiallyhyperbolic set ∆ ⊂ U for Ψ , the set | D | is a (global) c-section for ( Ψ , ∆ ) . Remark 3.10 (Fixed points nowhere dense) . This prompts us to note that the set offixed points of a partially hyperbolic dynamical system is nowhere dense: the set offixed points of a continuous dynamical system is closed, and the restriction to it is theidentity. The interior being nonempty is incompatible with partial hyperbolicity. Thusthe c -admissible disk family in Proposition 3.9 can be chosen away from the set of fixedpoints.The next step is a result about stable accessibility on center disks. Its proof needsadaptations for flows, the core part of which is Lemma 3.15 below. CCESSIBILITY AND CENTRALIZERS FOR FLOWS 7
Proposition 3.11 ([5, Proposition 1.3]) . If Λ is a partially hyperbolic set for a flow Φ and δ > , then (with the notations of Remark 2.2) there exist T > and a neighborhood U of Λ such that U ⊂ U ( Φ , Λ ) and if D is a c-admissible disk family with respect to ( Φ , Λ ) withr ( D ) < T − and R ( D ) > T , then for all σ > there exists Ψ ∈ U ( Φ , Λ ) such that:(1) d ( Φ , Ψ ) < δ ,(2) d ( Φ , Ψ ) < σ , and(3) if D ∈ D and ∆ ⊂ U is a bisaturated partially hyperbolic set for Ψ , then ( Ψ , ∆ ) isstably accessible on D,(4) if Φ preserves a volume, symplectic, or contact form, then so does Ψ . Theorem 3.1 follows from Propositions 3.9 and 3.11 just as [5, Theorem B] follows from[5, Propositions 1.3 and 1.4] in [5, Section 1.6], including the preservation of a volume,symplectic, or contact form.
Proof of Proposition 3.11.
Much of the proof of this proposition is exactly as in [5]. Wewill explain the ideas in these parts while highlighting the points that need modificationsfor flows.The first step [5, Lemma 2.1] introduces smaller disks that are sufficiently close, butdisjoint from, a c -admissible disk and have the property that there are su -paths atsufficiently small scales connecting any point in the c -admissible disk to one of thesmaller disks, not just for the original map, but also for any maps that are sufficiently C close.For these smaller disks one can then perform perturbations so that the flow is ac-cessible on them. Then one can show the perturbed flow will be accessible on D (the c -admissible disk for the original flow). By the choice of the c -admissible family weobtain accessibility of the bisaturated set for the perturbed flow.The existence of the smaller disks does not use perturbations and assumes the exis-tence of a c -admissible disk for a partially hyperbolic map. This holds in our setting byconsidering time- t maps. We include the statement of the result for completeness andadapt it for flows. Lemma 3.12.
There exist δ , ρ > , K > and a neighborhood U of Λ such that forany ρ ∈ (0, ρ ) , any c-admissible disk D with radius ρ , centered at p ∈ Λ , and for any (cid:178) ∈ (0, K − ρ ) , there exist z ,..., z (cid:96) ∈ T p M such that:(1) The balls B ( z i ,100 d (cid:178) ) are in the K (cid:178) -neighborhood of f − p ( D ) .(2) The balls B ( z i ,100 d (cid:178) ) are pairwise disjoint.(3) For any x ∈ D, there exists some z i such that for any Ψ that is δ -close to Φ in theC distance and for any bisaturated set ∆ ⊂ U for Ψ :(a) if x ∈ ∆ , then there is a su-path for Ψ between x and f p ( B ( z i , (cid:178) )) ,(b) if f p ( B ( z i , (cid:178) )) ⊂ ∆ , then any point y ∈ f p ( B ( x , (cid:178) /2)) belongs to an su-path thatintersects f p ( B ( z i , (cid:178) )) . The idea of the proof of Theorem 1.1 is to create small perturbations of the flow Φ supported near the points z ,..., z (cid:96) to create accessibility near each z i . This requires thefollowing notion (which will be used in Lemma 3.14 to show accessibility near the z i ). Definition 3.13 ( θ -accessibility) . A pair ( Ψ , ∆ ) of a flow and a bisaturated set is θ -accessible on f p ( B ( z ,2 d (cid:178) ) if there exist an orthonormal basis w ,..., w c of E cp and foreach j ∈ {1,..., c } a continuous map H j : [ − × [0,1] × f − p ( ∆ ) ∩ B ( z ,2 d (cid:178) ) → f − p ( ∆ ) ∩ B (0,2 ρ ) TODD FISHER AND BORIS HASSELBLATT such that for any x ∈ f − p ( ∆ ) ∩ B ( z ,2 d (cid:178) ) and s ∈ [ − H j ( s ,0, x ) = x ,(b) the map f p ◦ H j ( s ,., x ) : [0,1] → ∆ is a 4-legged su -path (Brin quadrilateral), i.e.,the concatenation of 4 curves, each contained in a stable or unstable leaf inalternation,(c) (cid:107) H j ( s ,1, x ) − x (cid:107) < (cid:178) d , and(d) (cid:107) H j ( ± x ) − ( x ± θ(cid:178) w j ) (cid:107) < θ (cid:178) d .The second step is that θ -accessibility in a neighborhood of a point implies accessi-bility on a smaller neighborhood. This is a restatement of [5, Lemma 2.2]. The proof isagain almost the same, but we provide it for completeness.From now on write d : = u + c + s , where dim E u = u , dim E c = c , and dim E s = s . Lemma 3.14.
For any θ > , there exist δ , ρ > and a neighborhood U of Λ such that(1) for any p ∈ Λ , any z ∈ B ( p , ρ ) ⊂ T p M and (cid:178) ∈ (0, ρ ) ,(2) for any flow Ψ that is δ -close to Φ in the C topology,(3) for any bi-saturated set ∆ ⊂ U such that ( Ψ , ∆ ) is θ -accessible on f p ( B ( z ,2 d (cid:178) ) ,the pair ( Ψ , ∆ ) is accessible on f p ( B ( z , (cid:178) )) .Proof. We let v ,..., v u be an orthonormal basis of E up and v u + c + ,..., v d an orthonormalbasis for E sp . We define local flows Φ i on f − p ( ∆ ) as follows: let X i be a vector field alongthe leaves of f − p ( W j Ψ ) where X i ( x ) = D π jx ( x + v i ) for j = u if 1 ≤ i ≤ u and j = s if u + c + ≤ j ≤ d , and the local flow is defined by the vector field X i on the set B (0,2 ρ ) ∩ f − p ( ∆ )and ρ is given by Lemma 3.12. So the orbit of x is the projection by π jx on the curve t (cid:55)→ x + t v i for | t | < ρ , and the orbits are C curves whose tangent space is arbitrarilyclose to (cid:82) v i for sufficiently small constants ρ , δ , and U as in Lemma 3.12.For ρ , δ , and U sufficiently small we see that(3) (cid:107) ϕ ti ( x ) − ( x + t θ(cid:178) v i ) (cid:107) < | t | θ (cid:178) d .We also let v u + j = w j be an orthonormal basis for the center direction and define induc-tively ϕ tu + j ( x ) = H j ( t ,1, x ) when t ∈ [0,1), ϕ tu + j ( x ) = ϕ t − u + j ◦ ϕ u + j ( x ) when t > ϕ tu + j ( x ) = ϕ t + u + j ◦ ϕ − u + j ( x ) when t < t so long as it can be defined. From properties (c) and (d) inthe definition of H j and estimate 3 above we let P ( t ,..., t d ) = ϕ t ... ϕ t d d ( x ) for ( t ,..., t d ) ∈ [ − θ − ,3 θ − ] d . This is a continuous map and (cid:107) P ( t ,..., t d ) − ( x + (cid:80) i t i θ(cid:178) v i ) (cid:107) < (cid:178) . Theimage of P contains B ( x , (cid:178) ), and f p ◦ P shows that ( Ψ , ∆ ) is accessible on B ( z , (cid:178) ). (cid:3) The adaptation to flows.
We next produce the perturbations near the z i , from Lemma3.12, that we need to establish accessibility. This result and proof are similar to [5, Lemma2.3], but in our case our perturbations need to be constructed for flows instead of maps.This is the essential adaptation of the Avila–Crovisier–Dolgopyat–Wilkinson arguments. Lemma 3.15.
Consider a partially hyperbolic flow Φ generated by a vector field X . Withthe previous notations, there exist η , α > such that for any α ∈ (0, α ) , p ∈ Λ , z ∈ B (0,1/4) ⊂ T p M with X ( f p ( z )) (cid:54)= , r ∈ (0,1/4) and any unit vector v ∈ E cp there is avector field Y such that: CCESSIBILITY AND CENTRALIZERS FOR FLOWS 9 (1) Y = X outside f p ( B ( z ,3 r )) ,(2) d f − p Y = d f − p X + αη v on B ( z ,2 r ) ,(3) Y is α -close to X ,(4) the flow Ψ defined by Y is r d -close to Φ in the C distance.(5) if Φ preserves a volume, symplectic, or contact form, then so does Ψ .Proof. Figure 1 (which utilizes that by Remark 3.10 we are working away from fixedpoints) illustrates what we would like to achieve: to connect the perturbed vector field (inthe smallest circle) to the original one (in the square) by a bump-function interpolation.For arbitrary vector fields this is all there is. For volume-preserving flows, we invokeF
IGURE
1. The perturbations in Lemma 3.15the pasting lemma [35, Theorem 1] (see also [3]) to ensure volume-preservation of theperturbation.For symplectic flows take symplectic flow-box charts as in Proposition 3.2 in the rectan-gle and large circle in Figure 1; they are locally Hamiltonian with constant Hamiltonianson each neighborhood, so we can interpolate the Hamiltonians in the annulus.For contact flows take a Darboux chart (Remark 3.4) on the rectangle, then put a suit-ably rotated and scaled version in the large disk. This defines a local contact form whoseReeb field is as desired; interpolate the contact forms in the annulus. (For perturbationsin the flow direction, i.e., reparameterization, no rotation is needed.) (cid:3)
To construct the desired flow we will use the above perturbation to establish θ -accessibility for the sets f p i ( B ( z i ,2 d (cid:178) )). We do this by adjusting Brin quadrilateralsby using the perturbation above. Before explaining this step we first adjust the neighbor-hood U and describe the setup we will need.We first let C s and C u be cone fields in U that are Φ invariant for t < t > δ we know that the cone fields are invariantfor any flow that is δ -close in the C topology to Φ . For T > U be a neighborhoodof Λ such that(4) U ⊂ U ∩ U ∩ (cid:92) | t |≤ T ϕ t ( U ).We also define C u = D ϕ T ( C u ) and C s = D ϕ T ( C s ) on U . We know there exists some T > ρ > T ≥ T and ρ < ρ , then for any p ∈ Λ we have f p ( B (0,2 ρ )) ⊂ U and the cone fields D f − p ( C s ) and D f − p ( C u ) on B (0,2 ρ ) are γ -close to E sp and E up in T p M where γ is smaller than αη .Let ρ ∈ (0,min{ ρ , ρ , ρ , ρ }) and fix T ≥ T so that any c -admissible disk D withcenter p ∈ Λ and r ( D ) < T − satisfies f p ( D ) ⊂ B (0, ρ ) ⊂ T p M . We also have U defined by T satisfying (4) and the existence of a family D of c -admissible disks from Proposition3.9 for some σ > θ = αη d . We now explain the quadrilaterals we will use to establish θ -accessibilityfor the perturbed flow. For D ∈ D we fix the z i as in Lemma 3.12 and the sets f p i ( B ( z i ,100 d (cid:178) )) ⊂ U for (cid:178) sufficiently small. We can also define subspaces E s and E u such that • D f p i ( z i ) E s ⊂ C s ( f p i ( z i )), • D f p i ( z i ) E u ⊂ C u ( f p i ( z i )), • dim E s = dim E sp i , and • dim E u = dim E up i .Let v s ∈ E s and v u ∈ E u be unit vectors and fix w ,..., w c an orthonormal basis for E cp i . For the foliations F u and F s we define the flows Φ (cid:48) k that corresponds to the linearflow ( x , t ) (cid:55)→ x + t v k projected to the leaves of F k for k ∈ { u , s }. For each j ∈ {1,..., c } weexamine the quadrilaterals given by the composition v i , j = ϕ (cid:48) s ( − j d (cid:178) ) ◦ ϕ (cid:48) u ( − d (cid:178) ) ◦ ϕ (cid:48) s (10 j d (cid:178) ) ◦ ϕ (cid:48) u (10 d (cid:178) )and v i , − j = ϕ (cid:48) s (10 j d (cid:178) ) ◦ ϕ (cid:48) u (10 d (cid:178) ) ◦ ϕ (cid:48) s ( − j d (cid:178) ) ◦ ϕ (cid:48) u ( − d (cid:178) ).We define R ( i , (cid:178) ) : = max ≤ j ≤ c (cid:161) max( (cid:107) v i , j (cid:107) , (cid:107) v i , − j (cid:107) ) (cid:162) .Because R ( i , (cid:178) ) ∈ o ( (cid:178) ) [20, equation (8)], we have: Proposition 3.16.
For each z i there exists some (cid:178) > such that for (cid:178) ∈ (0, (cid:178) ) we haveR ( i , (cid:178) ) < θ(cid:178) /10 d .We let Ψ be the flow generated by a vector field whose restriction to B ( z i + j d (cid:178) v s ,2 d (cid:178) )and B ( z i − j d (cid:178) v s ,2 d (cid:178) ) satisfy the conditions in Lemma 3.15.Then starting at z i and using the quadrilaterals above we see that the the new flowcoincides with translation by θ(cid:178) w j from the original flow. Indeed by construction firstleg of the quadrilateral is left unperturbed by the new flow. Similarly, the second leg is leftunperturbed. The third leg is the composition of x (cid:55)→ x − (10 d (cid:178) ) v u with the translation θ (cid:178) w j . The fourth leg similarly corresponds with the composition with the linear flow andtranslation by θ (cid:178) w j . Then the quadrilateral on B ( z i ,2 d (cid:178) ) corresponds with translationby θ(cid:178) w j , see Figure 2.Similarly, the quadrilateral associated with − j corresponds with translation by − θ(cid:178) w j from the original flow. From this we obtain the desired function H j used to ensure θ -accessibility, see Figure 3.Furthermore, from the size of the perturbations we see that the flows are at least (cid:178) d -close in the C topology. This is the desired flow to prove Proposition 3.11.Let ∆ ⊂ U be a bisaturated set for Ψ . Let (cid:101) Ψ be C -close to Ψ and ˜ ∆ ⊂ U be a bisaturatedset for (cid:101) Ψ contained in a small neighborhood of ∆ . By the construction above we knowthat ( (cid:101) Ψ , (cid:101) ∆ ) is θ -accessible on each of the f p i ( B ( z i ,2 d (cid:178) )). Then ( (cid:101) Ψ , (cid:101) ∆ ) is accessible oneach of the sets f p i ( z i , (cid:178) ).For D ∈ D that intersects (cid:101) ∆ at a point z we know there exists a su -path for (cid:101) Ψ from z to a point y ∈ f p i ( B ( z i , (cid:178) )) for some i from Lemma 3.12. Furthermore, from Lemma CCESSIBILITY AND CENTRALIZERS FOR FLOWS 11 F IGURE
2. Perturbed quadrilateralF
IGURE θ -accessibility3.12 we know if x is (cid:178) /2 close to z in D , then there is a su -path to a point y (cid:48) ∈ f p i ( B ( z i , (cid:178) )).Accessibility on B ( z i , (cid:178) ) implies there is a su -path from y to y (cid:48) . So any point in the (cid:178) /2-neighborhood of z contained in D is in the same accessibility class as z . This implies thatevery point in D is in the same accessibility class for ( (cid:101) Ψ , (cid:101) ∆ ) since D is connected. Then( Ψ , ∆ ) is stably accessible on any disk D ∈ D . (cid:3)
4. C
ENTRALIZERS
With Theorem 1.1 in hand, we now adapt arguments by Burslem from the proof ofTheorem 2.5 [16, Lemmas 5.2, 5.3] in order to prove our Theorem 1.2. We will use the following criterion, which holds for flows such that t (cid:55)→ ϕ t is invertible near t = Proposition 4.1.
A partially hyperbolic flow Φ has trivial flow-centralizer (Definition1.5) if Φ has discrete diffeomorphism-centralizer, i.e., if for any f : M → M, which is C ,commutes with Φ and is sufficiently close to the identity, there is a τ near 0 such thatf = ϕ τ .Proof. If a flow Ψ commutes with Φ then discreteness of the centralizer implies that ψ s = ϕ τ ( s ) for small enough s , hence also that id = ψ = ϕ τ (0) , i.e., τ (0) = Φ is notperiodic).Since D ϕ τ (0) ( p ) = id = D ψ ( p ) = lim s → D ψ s ( p ) = lim s → D ϕ τ ( s ) ( p ) (with p as above),partial hyperbolicity implies that τ is continuous at 0, and the commutation relationgives additivity : τ ( s + t ) = τ ( s ) + τ ( t ).Together, these give a c ∈ (cid:82) such that τ ( s ) = cs for all t ∈ (cid:82) : Continuity at 0 impliescontinuity at any s because τ ( s + (cid:178) ) = τ ( s ) + τ ( (cid:178) ) −−−− (cid:178) → → τ ( s ); additivity implies linearity of τ on (cid:81) , then continuity implies linearity on (cid:82) [2, Theorem 1, §2.1], [1, §2.1.1]. (cid:3) We note that assuming the absence of fixed points makes a few of the argumentsbelow a little simpler.
Proof of Theorem 1.2.
First, we produce a (nonfixed) closed orbit that is isolated amongclosed orbits of at most twice its period.There is a nonwandering point that is not fixed: every point in the support of aninvariant Borel probability measure is nonwandering, and a partially hyperbolic flowhas a nonatomic ergodic invariant measure because the topological entropy is positive[17, Theorem 2] while atomic measures have zero entropy. From this nonwandering point, the Pugh Closing Lemma gives a C -close (partiallyhyperbolic) flow with a closed orbit, and this works for volume-preserving or symplecticflows [4].Contact flows always have a closed orbit by the Weinstein Conjecture [34].By the Transversality Theorem [25, Theorems A.3.19, 7.2.4 & p. 296], this closed orbitcan (for a C r -open C -dense set of such flows) be taken transverse and hence isolatedamong closed orbits of up to twice its period.Theorem 1.1 then gives a C -open dense set of accessible (volume-preserving/sym-plectic/contact) partially hyperbolic flows Φ with a transverse closed orbit O ( p ) that is(hence) isolated among closed orbits of at most twice its period.We now verify that their diffeomorphism-centralizer is discrete (Proposition 4.1).If f : M → M is C , commutes with Φ and is sufficiently close to the identity, then f maps closed orbits of Φ to closed orbits of Φ with the same period. Thus, since O ( p )is isolated among orbits of the same period, f ( p ) ∈ O ( p ) once f is sufficiently close tothe identity, and indeed f ( p ) = ϕ τ ( p ) for some τ near 0. Thus h : = f ◦ ϕ − τ fixes p andcommutes with Φ . Here, “sufficiently close” means that d C ( h ,id) < (cid:178) , where (cid:178) > h = id, and it suffices to verify this on the denseset M of points that are accessible from p with ( su )-paths that avoid fixed stable andunstable leaves, i.e., disjoint from W u ( x ) and W s ( x ) for any fixed point x . (This set is Lemma 4.4 below bypasses this entropy argument—which is not needed when volume is preserved—atthe expense of an initial perturbation.
CCESSIBILITY AND CENTRALIZERS FOR FLOWS 13 dense because the set of fixed points is finite and the invariant manifolds have positivecodimension.)For any y ∈ M , recursive application of Lemma 4.2 below to a finite ( su)-path from p to y shows that h fixes all vertices of this path and hence y . (cid:3) Lemma 4.2 ([16, Lemma 5.3]) . If h ( q ) = q and W u ( q ) contains no fixed point, thenh ( x ) = x for all x ∈ W u ( q ) . Likewise for W s ( q ) .Proof. Suppose x ∈ W u ( q ); the case x ∈ W s ( q ) is analogous. Then h ( x ) ∈ h ( W u ( q )) h ∈ C ===== W u ( h ( q )) h ( q ) = q ====== W u ( q ) = W u ( x ),while d ( ϕ t ( x ), ϕ t ( h ( x ))) = d ( ϕ t ( x ), h ( ϕ t ( x ))) < d C ( h ,id) < (cid:178) .This implies h ( x ) = x by Lemma 4.3 below. (cid:3) The proof of [16, Lemma 5.2] applies to leaves containing no fixed points:
Lemma 4.3.
There is an (cid:178) > such that if W u ( x ) contains no fixed point and y ∈ W u ( x ) satisfies d ( ϕ t ( x ), ϕ t ( y )) < (cid:178) for all t ∈ (cid:82) , then x = y. Likewise for W s ( x ) . As promised, we provide here an alternate argument for the existence of the nonfixednonwandering point.
Lemma 4.4.
A partially hyperbolic flow can be C -perturbed to have a nonwanderingpoint that is not fixed.Proof. For volume-preserving flows, all points are nonwandering by the Poincaré Recur-rence Theorem, so there are nonwandering points which are not fixed.Without volume-preservation, the Kupka–Smale Theorem [21, Theorem 6.1.6] givesa C r perturbation for which all fixed points are hyperbolic (and hence also finite innumber). Unless there is an additional nonwandering point, the nonwandering setis then hyperbolic and equal to the limit set [21, Proposition 1.5.17]—so the union oftheir (finitely many!) stable manifolds is M [21, Proposition 5.3.40], contrary to partialhyperbolicity (stable manifolds have positive codimension). (cid:3) R EFERENCES[1] J. Aczél.
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CCESSIBILITY AND CENTRALIZERS FOR FLOWS 15
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ISHER , D
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