The Katok-Spatzier Conjecture and Generalized Symmetries
aa r X i v : . [ m a t h . D S ] A ug THE KATOK-SPATZIER CONJECTURE AND GENERALIZEDSYMMETRIES
LENNARD F. BAKKER
Abstract.
Within the smooth category, an intertwining is exhibited between theglobal rigidity of irreducible higher-rank Z d Anosov actions on T n and the classifica-tion of equilibrium-free flows on T n that possess nontrivial generalized symmetries. Introduction
In [10], Spatzier communicated the conjecture that any irreducible higher-rank Z d Anosov action is C ∞ -conjugate to an algebraic action. Later, in [16], Kalinin and Spatzierstated a refinement of this conjecture that contends that any irreducible higher-rank Z d Anosov action on any compact manifold has a finite cover C ∞ -conjugate to an algebraicaction. The asserted global rigidity was motivated in part by earlier results of Katok andLewis in [14] and [15], and a more recent result by Rodriguez Hertz in [21]. In the latter,global rigidity has been shown for any higher-rank Z d Anosov action on T n whose actionon homology has simple eigenvalues and whose course Lyapunov spaces are one or twodimensional, plus additional conditions. A partial confirmation of the refined assertionof global rigidity is provided in [16] for higher-rank Z d Anosov C ∞ actions each of whosecourse Lyapunov spaces are one-dimensional, plus additional conditions. If a higher-rank Z d Anosov action on T n is C ∞ -conjugate to an algebraic one and that algebraicaction has a common real eigenvector, then that higher-rank Z d Anosov action preservesa one-dimensional C ∞ foliation of T n determined by that common real eigenvector, i.e.,generated by a equilibrium-free C ∞ flow.This paper intertwines the global rigidity of higher-rank Z d Anosov C ∞ actions on T n with the classification of equilibrium-free C ∞ flows on T n that possess nontrivial gen-eralized symmetries. The intertwining centers on presence of a single one-dimensional C ∞ distribution determined by an equilibrium-free C ∞ flow that is invariant under a Z d Anosov C ∞ action, without a priori conditions on all the course Lyapunov spaces. Asshown in Section 3, any generalized symmetry of an equilibrium-free flow is nontrivial ifit is Anosov (see Corollary 3.2). Furthermore, any equilibrium-free flow that possesses anontrivial generalized symmetry does not have any uniformly hyperbolic compact invari-ant sets (see Corollary 2.3). The intertwining juxtaposes an equilibrium-free C ∞ flowthat is not Anosov with a smooth Z d action that is Anosov. In the C ∞ topology, this isa counterpoint to the result of Palis and Yoccoz in [20] on the triviality of centralizersfor an open and dense subset of Anosov diffeomorphisms on T n , and also to the result ofSad in [22] on the local triviality of centralizers for an open and dense subset of AxiomA vector fields that satisfy the strong transversality condition (as applied to vector fieldson T n ). In particular, it is quite rare for an equilibrium-free C ∞ flow on T n (or more Mathematics Subject Classification.
Primary: 37C55, 37C85; Secondary: 11R99.
Key words and phrases.
Anosov Actions, Generalized Symmetries, Equilibrium-free Flows. Spatzier’s Conjecture and Generalized Symmetries generally, on a closed Riemannian manifold) to possess a nontrivial generalized symmetry(see Corollary 2.2).The first aspect of the intertwining on T n relates the global rigidity of a Z d Anosov C ∞ action with an equilibrium-free C ∞ flow that is quasiperiodic. As detailed in Section2, the generalized symmetry group of a C ∞ flow Φ is the subgroup S Φ of Diff ∞ ( T n ) eachof whose elements R sends (via the pushforward) the generating vector field X Φ of Φ toa uniform scalar multiple ρ Φ ( R ) of itself. The multiplier group M Φ of Φ is the abeliangroup of these scalars. As shown in Section 4, the elements of M Φ \ { , − } when Φ isquasiperiodic (or more generally, minimal) are algebraic integers of degree between 2 and n inclusively (see Corollary 4.3). Theorem 1.1. On T n , suppose α is a Z d Anosov C ∞ action, and Φ is equilibrium-free C ∞ flow. If α ( Z d ) ⊂ S Φ and Φ is quasiperiodic ( i.e., C ∞ -conjugate to an irrationalflow ) , then α ( Z d ) is C ∞ -conjugate to an affine action, a finite index subgroup of α ( Z d ) is C ∞ -conjugate to an algebraic action, and M Φ contains a Z d subgroup. Relevant definitions and the proof are given in Section 6. The proof holds not only for d ≥ d = 1. It uses a semidirect product characterization of the structureof the generalized symmetry group for an irrational flow (as shown in Section 5). It alsouses the existence of a common fixed point for a finite index subgroup of the Z d Anosovaction, a device used in other global rigidity results (for example, see [15]).The second aspect of the intertwining on T n relates the classification of an equilibrium-free C ∞ flow with a Z d Anosov C ∞ action that is topologically irreducible. As detailedin Section 7, an irrational flow φ on T n is of Koch type if a uniform scalar multiple ofits frequencies form a Q -basis for a real algebraic number field of degree n (also see [17]and [18]). For an R ∈ S Φ of an equilibrium-free C ∞ flow Φ, the quantity log | ρ Φ ( R ) | isthe value of the Lyapunov exponent χ R of R in the direction of X Φ (see Theorem 3.1). Theorem 1.2. On T n , suppose α is a higher-rank Z d Anosov C ∞ action, and Φ isequilibrium-free C ∞ flow. If α ( Z d ) ⊂ S Φ , and α is topologically irreducible and C ∞ -conjugate to an algebraic Z d action, and for an Anosov element R ∈ α ( Z d ) , the multi-plicity of the value log | ρ Φ ( R ) | of χ R is one at some point of T n , then Φ is projectively C ∞ -conjugate to an irrational flow of Koch type. Relevant definitions and the proof are given in Section 7. The proof uses the Oseledetsdecomposition for an Anosov diffeomorphism (see [6] and [12]) to show that the flowis C ∞ -conjugate to one generated by a constant vector field. Then by the topologicalirreducibility and results of Wallace in [25], the components of a scalar multiple of theconstant vector field are shown to form a Q -basis for a real algebraic number field.2. Flows with Nontrivial Generalized Symmetries
Generalized symmetries extend the classical notions of time-preserving and time-reversing symmetries of flows. To simplify notations for these and for proofs of results, itis assumed throughout the remainder of the paper that all manifolds, flows, vector fields,diffeomorphisms, distributions, etc., are smooth, i.e., of class C ∞ . Let P be a closed(i.e., compact without bounday) manifold. Let Flow( P ) denote the set of flows on P .Following [5], a generalized symmetry of ψ ∈ Flow( P ) is an R ∈ Diff( P ) such that thereis µ ∈ R × = R \ { } (the multiplicative real group) for which Rψ ( t, p ) = ψ ( µt, R ( p )) for all t ∈ R and all p ∈ P. .F. Bakker 3 It is easy to show that R being a generalized symmetry of ψ is equivalent to R satisfying R ∗ X ψ = µX ψ for some µ ∈ R × . Here X ψ ( p ) = ( d/dt ) ψ ( t, p ) | t =0 is the vector field that generates ψ , and R ∗ X ψ = T RX ψ R − is the push-forward of X ψ by R where T R is the derivative map. The generalized symmetry group of ψ is the set S ψ of all the generalized symmetries that ψ possesses. There is a homomorphism ρ ψ : S ψ → R × taking R ∈ S ψ to its uniquemultiplier ρ ψ ( R ) = µ . The multiplier group of ψ is M ψ = ρ ψ ( S ψ ).The generalized symmetry group and the multiplier group of a flow are invariants forthe equivalence relation of projective conjugacy. Two ψ, φ ∈ Flow( P ) are projectivelyconjugate if there are h ∈ Diff( P ) and ϑ ∈ R × such that h ∗ X ψ = ϑX φ . Projectiveconjugacy is an equivalence relation on Flow( P ). Projective conjugacy reduces to smoothconjugacy when ϑ = 1. For h ∈ Diff( P ), let ∆ h be the inner automorphism of Diff( P )given by ∆ h ( R ) = h − Rh for R ∈ Diff( P ). If h ∗ X ψ = ϑX φ , then ∆ h ( S φ ) = S ψ (seeTheorem 4.1 in [5] which states that S ψ is conjugate to the generalized symmetry groupfor the flow determined by ϑX φ , which is exactly the same as S φ .) Furthermore, if ψ and φ are projectively conjugate, then M ψ = M φ (see Theorem 2.2 in [3]), i.e., the multipliergroup is an absolute invariant of projective conjugacy.Any R ∈ S ψ is a trivial generalized symmetry of ψ if ρ ψ ( R ) = 1 (i.e., R is time-preserving), or if ρ ψ ( R ) = − R is time-reversing). Any R ∈ S ψ with | ρ ψ ( R ) | 6 = 1 isa nontrivial generalized symmetry of ψ . A flow ψ (or equivalently its generating vectorfield X ψ ) is said to possess a nontrivial generalized symmetry when M ψ \ { , − } 6 = ∅ . Theorem 2.1.
Let ψ be a flow on a closed Riemannian manifold P . If ψ has a periodicorbit and M ψ \ { , − } 6 = ∅ , then ψ has a nonhyperbolic equilibrium.Proof. Suppose for p ∈ P that O ψ ( p ) = { ψ t ( p ) : t ∈ R } is a periodic orbit whosefundamental period is T >
0. (Here ψ t ( p ) = ψ ( t, p ).) Assuming that M φ \ { , − } 6 = ∅ implies there is Q ∈ S ψ such that ρ ψ ( Q ) = ±
1. If | ρ ψ ( Q ) | >
1, then | ρ ψ ( Q − ) | <
1, since ρ ψ is a homomorphism. Hence there is R ∈ S ψ such that | ρ ψ ( R ) | <
1. Then R ( p ) = Rψ ( T, p ) = ψ (cid:0) ρ ψ ( R ) T , R ( p ) (cid:1) . For p = R ( p ), this implies that O ψ ( p ) is a periodic orbit with period T = | ρ ψ ( R ) | T .Suppose that T is not a fundamental period for O ψ ( p ), i.e., there is 0 < T ′ < T suchthat p = ψ ( T ′ , p ). This implies that R ( p ) = p = ψ ( T ′ , p ) = ψ (cid:0) T ′ , R ( p ) (cid:1) = Rψ ( T ′ /ρ ψ ( R ) , p ) . Since R is invertible, this gives p = ψ ( T ′ /ρ ψ ( R ) , p ). With T as the fundamental periodfor O ψ ( p ), there is m ∈ Z + such that | T ′ /ρ ψ ( R ) | = mT . Hence T ′ = mT | ρ ψ ( R ) | ≥ T | ρ ψ ( R ) | = T , a contradiction to T ′ < T . Thus, T is the fundamental period for O ψ ( p ). Since | ρ ψ ( R ) | <
1, it follows that O ψ ( p ) and O ψ ( p ) have different fundamental periods, andhence are distinct periodic orbits. Iteration gives a sequence of distinct periodic orbits O ψ ( p i ), where p i = R i ( p ), whose fundamental periods T i = | ρ ψ ( R ) | i T decrease to 0since | ρ ψ ( R ) | < l ( p i ) be the arc length of O ψ ( p i ). In terms of the Riemannian norm k · k on T P ,the tangent bundle of P , l ( p i ) = Z T i k X ψ (cid:0) ψ ( t, p i ) (cid:1) k dt. Spatzier’s Conjecture and Generalized Symmetries
Continuity of X ψ on the compact manifold P implies that M = sup p ∈ P k X ψ ( p ) k is finite.Thus l ( p i ) ≤ M T i → . By the compactness of P , there is a convergence subsequence p i m with limit p ∞ . If k X ψ ( p ∞ ) k 6 = 0, then the smoothness of X ψ implies by the Flow Box Theorem that thereare no periodic orbits of ψ completely contained in a sufficiently small neighborhood U of p ∞ . But since l ( p i m ) → p i m → p ∞ , there are periodic orbits of ψ completelycontained in U , a contradiction. Thus p ∞ is an equilibrium for ψ . If p ∞ were hyperbolic,then the Hartman-Grobman Theorem would imply that there are no periodic orbitscompletely contained in a sufficiently small neighborhood of p ∞ , again a contradiction.Therefore, φ has a nonhyperbolic equilibrium. (cid:3) The flows on a closed Riemannian manifold which possess nontrivial generalized sym-metries are rare or non-generic in the sense of Baire category theory. Let X ( P ) denotethe set of vector fields on P . With P compact, there is a one-to-one correspondencebetween X ( P ) and Flow( P ). Equipped with the usual C ∞ topology, X ( P ) is a Bairespace. A residual subset of X ( P ) is one that contains a countable intersection of open,dense subsets of X ( P ). Because X ( P ) is a Baire space, any residual subset is dense. Corollary 2.2.
For a closed Riemannian manifold P , the set of vector fields on P whichdo not possess nontrivial generalized symmetries is a residual subset of X ( P ) .Proof. Let V be the set of X in X ( P ) such that any equilibrium, if any, of the flow ψ induced by X is hyperbolic, and there is at least one periodic orbit for ψ . By Theorem2.1, none of the vector fields in V possess nontrivial generalized symmetries. Let H bethe subset of X in X ( P ) such that the flow ψ induced by X has periodic orbits all ofwhich are hyperbolic, and any equilibrium it has is hyperbolic. By definition, H ⊂ V ,and by the Kupka-Smale Theorem, H is a residual subset of X ( P ). (cid:3) According to Theorem 2.1, flows which might possess nontrivial generalized symme-tries are those that are equilibrium-free and without periodic orbits. The possession ofa nontrivial generalized symmetry for an equilibrium-free flow places dynamical restric-tions on the compact invariant sets of the flow. An invariant set Λ for an equilibrium-free Φ ∈ Flow( P ) is uniformly hyperbolic if there is a continuous T Φ-invariant splitting T p Λ = E s ( p ) ⊕ Span( X Φ ( p )) ⊕ E u ( p ) for p ∈ Λ and constants C ≥ λ ∈ (0 ,
1) suchthat k T p Φ t ( v ) k ≤ Cλ t k v k for v ∈ E s ( p ) and t ≥ k T p Φ − t ( v ) k ≤ Cλ t k v k for v ∈ E u ( p ) and t ≥ . Recall that if Λ = P is uniformly hyperbolic for Φ, then Φ is called Anosov. Corollary 2.3. If Φ is an equilibrium-free flow on a closed Riemannian manifold P which possesses a nontrivial generalized symmetry, then any compact invariant set for Φ is not uniformly hyperbolic ( and, in particular, Φ is not Anosov ) .Proof. Suppose Λ is a uniformly hyperbolic compact invariant set of Φ. By the AnosovClosing Lemma, there exists a periodic orbit for Φ. But with Φ being equilibrium-freeand possessing a nontrivial generalized symmetry, Theorem 2.1 implies that Φ does nothave periodic orbits. Thus Λ can not be uniformly hyperbolic. (cid:3) .F. Bakker 5 Multipliers and Lyapunov Exponents
The multiplier of a generalized symmetry for an equilibrium-free flow is related to theLyapunov exponents of that generalized symmetry. For a closed Riemannian manifold P , the Lyapunov exponent of R ∈ Diff( P ) at p ∈ P is χ R ( p, v ) = lim sup m →∞ log k T p R m ( v ) k m , v ∈ T p P \ { } . It is independent of the Riemannian norm on P because P is compact. As the follow-ing result shows, any trivial generalized symmetry for an equilibrium-free flow has zeroLyapunov exponents everywhere. Theorem 3.1. If ψ is an equilibrium-free flow on a closed Riemannian manifold P ,then for each R ∈ S ψ , the one-dimensional distribution E ⊂ T P determined by E ( p ) =Span (cid:0) X ψ ( p ) (cid:1) is R -invariant and χ R (cid:0) p, X ψ ( p ) (cid:1) = log | ρ ψ ( R ) | for all p ∈ P. Proof.
Each R ∈ S ψ satisfies T p R (cid:0) X ψ ( R − ( p ) (cid:1) = ρ ψ ( R ) X ψ ( p ) for all p ∈ P . Then T p R ( X ψ ( p )) = ρ ψ ( R ) X ψ ( R ( p )). The distribution E ⊂ T P determined by E ( p ) =Span (cid:0) X ψ ( p ) (cid:1) is R -invariant and one-dimensional because ρ ψ ( R ) = 0 and X ψ ( p ) = 0 forall p ∈ P . Furthermore, it follows for all m ≥ T p R m ( X ψ ( p )) = [ ρ ψ ( R )] m X ψ (cid:0) R m ( p ) (cid:1) , and so k T p R m ( X ψ ( p )) k = | ρ ψ ( R ) | m k X ψ (cid:0) R m ( p ) (cid:1) k . The flow ψ being smooth and equilibrium-free on the compact manifold P implies that k X ψ (cid:0) R m ( p ) (cid:1) k is both bounded away from 0 and bounded above uniformly in m for each p ∈ P . Therefore χ R (cid:0) p, X ψ ( p ) (cid:1) = lim sup m →∞ m log | ρ ψ ( R ) | + log k X ψ (cid:0) R m ( p ) (cid:1) k m = log | ρ ψ ( R ) | for all p ∈ P . (cid:3) Global uniform hyperbolicity is a dynamical condition on a generalized symmetry ofan equilibrium-free flow that guarantees that it is nontrivial. An R ∈ Diff( P ) is Anosov if there is a continuous T R -invariant splitting T p P = E s ( p ) ⊕ E u ( p ) for p ∈ P , andconstants c > λ ∈ (0 ,
1) independent of p ∈ P , such that k T p R m ( v ) k ≤ cλ m k v k for v ∈ E s ( p ) and m ≥ , and k T p R − m ( v ) k ≤ cλ m k v k for v ∈ E u ( p ) and m ≥ E s ( p ) and E u ( p ) bounded away from 0 (see [6]). Theorem 3.2.
Let ψ be an equilibrium-free flow on a closed Riemannian manifold P .If R ∈ S ψ is Anosov, then | ρ ψ ( R ) | 6 = 1 .Proof. Suppose R ∈ S ψ is Anosov and | ρ ψ ( R ) | = 1. If ρ ψ ( R ) = −
1, then replacing R with R gives ρ ψ ( R ) = 1 with R Anosov. Let E s ( p ) ⊕ E u ( p ) be the continuous T R -invariantsplitting with its associated contraction estimates. The generating vector field for ψ hasa continuous decomposition X ψ ( p ) = v s ( p ) + v u ( p ) for v s ( p ) ∈ E s ( p ) and v u ( p ) ∈ E u ( p ).From the contraction estimates of T p R on E s ( p ) and E u ( p ) it follows for all p ∈ P that k T p R m ( v s ( p )) k → , k T p R − m ( v u ( p )) k → m → ∞ . Spatzier’s Conjecture and Generalized Symmetries
With ρ ψ ( R ) = 1 and X ψ ( p ) = v s ( p ) + v u ( p ), the equation R ∗ X ψ = ρ ψ ( R ) X ψ becomes T p R m ( v s ( p ) + v u ( p )) = X ψ ( R m ( p )) = v s ( R m ( p )) + v u ( R m ( p ))for all p ∈ P and m ∈ Z . The T R -invariance of E u ( p ) implies that T p R m ( v u ( p )) = v u ( R m ( p )) . For a fixed p ∈ P , there is by the compactness of P , a subsequence R m i ( p ) convergingto a point, say p ∞ , as m i → ∞ . Hence X ψ ( p ∞ ) = lim i →∞ X ψ ( R m i ( p ))= lim i →∞ T p R m i ( v s ( p ) + v u ( p ))= lim i →∞ (cid:2) T p R m i ( v s ( p )) + v u ( R m i ( p )) (cid:3) = v u ( p ∞ ) . It now follows thatlim m →∞ X ψ ( R − m ( p ∞ )) = lim m →∞ T p ∞ R − m ( X ψ ( p ∞ )) = lim m →∞ T p ∞ R − m ( v u ( p ∞ )) = 0 . Compactness of P implies that X ψ has a zero. But ψ is an equilibrium-free flow, andtherefore | ρ ψ ( R ) | 6 = 1. (cid:3) Any equilibrium-free flow with a generalized symmetry that is Anosov does not haveany periodic orbits according to Lemma 2.1 and Theorem 3.2. On the other hand, foran equilibrium-free flow that is without nontrivial generalized symmetries, Theorem 3.2implies that none of its generalized symmetries can be Anosov. However, the converse ofTheorem 3.2 is false. As illustrated next, a partially hyperbolic diffeomorphism can bea nontrivial generalized symmetry of an equilibrium-free flow without periodic orbits.
Example 3.3.
Let P = T n = R n / Z n , the n -torus, equipped with global coordinates( θ , θ , . . . , θ n ). The flow ψ generated by X ψ = ∂∂θ + ∂∂θ + · · · + ∂∂θ n − + ∂∂θ n − + √ ∂∂θ n is equilibrium-free and without periodic orbits. Let R ∈ Diff( T n ) be induced by theGL( n, Z ) matrix B = . . . . . . . . . . . . . . . , that is, T R = B . This R is a nontrivial generalized symmetry of ψ because R ∗ X ψ =(1 + √ X ψ , i.e., ρ ψ ( R ) = 1 + √
2. However B has an eigenvalue of 1 of multiplicity n −
2, and the eigenspace corresponding to this eigenvalue is an ( n − R . The other two eigenvalues of B are 1 ± √ R respectively.Thus R is partially hyperbolic. .F. Bakker 7 Restrictions on Multipliers
Algebraic restrictions may occur on ρ Φ ( R ) for a nontrivial generalized symmetry R ofan equilibrium-free flow Φ on a compact manifold P without boundary. This happenswhen there are interactions beyond R ∗ X Φ = ρ Φ ( R ) X Φ between the dynamics of R and Φon submanifolds diffeomorphic to T k for some 2 ≤ k ≤ dim( P ). A real algebraic integer is the root of a monic polynomial with integer coefficients, and its degree is the degreeof its minimal polynomial. As shown by Wilson in [26], there is for any integer k with2 ≤ k ≤ dim( P ) −
2, an equilibrium-free flow Φ on P which has an invariant submanifold N diffeomorphic to T k on which Φ is minimal , i.e., every orbit of Φ in N is dense in N .The well-known prototype of a minimal flow on T k is an irrational flow , i.e., a ψ on T k for which X ψ is a constant vector field whose components (or frequencies) are rationallyindependent (see [9]), i.e., linearly independent over Q . According to Basener in [7], anyminimal flow on T n is topologically conjugate to an irrational flow. Lemma 4.1.
If a flow φ on a compact manifold N without boundary is topologicallyconjugate to a flow ψ on T k ( k = dim( N )) and ψ is minimal, then M φ ⊂ M ψ and each µ ∈ M φ \ { , − } is an algebraic integer of degree between and k inclusively.Proof. Suppose there is a homeomorphism h : N → T k such that hφ t = ψ t h for all t ∈ R .Without loss of generality, it is assumed that ψ is an irrational flow, since any minimalflow on T k is topologically conjugate to an irrational flow.Let V ∈ S φ and set µ = ρ φ ( V ). In terms of the homeomorphism Q = hV h − on T k ,the multiplier µ passes through h to ψ : Qψ ( t, θ ) = hRφ ( t, h − ( θ )) = hφ ( µt, Rh − ( θ )) = ψ ( µt, Q ( θ )) , for all t ∈ R , θ ∈ T k , i.e., Qψ t = ψ µt Q . This does not say yet that Q is a generalized symmetry of ψ withmultiplier µ because Q is only a homeomorphism at the moment.Following [1], the homeomorphism Q of T k lifts uniquely to ˆ Q ( x ) = ˆ L ( x ) + ˆ U ( x ) + ˆ c on R k , i.e., π ˆ Q = Qπ where π : R k → T k is the covering map. Here the linear part ofthis lift is ˆ L ( x ) = Bx for B ∈ GL( k, Z ); the periodic part is ˆ U ( x ), i.e., ˆ U ( x + ν ) = ˆ U ( x )for all x ∈ R k and all ν ∈ Z k , is continuous and satisfies ˆ U (0) = 0; and the constant partis ˆ c ∈ [0 , k . A lift of the irrational flow ψ to R n is ˆ ψ ( t, x ) = x + td where d = X ψ and x ∈ R k . For all t ∈ R , a lift of Qψ t to R k is ˆ Q ˆ ψ t and a lift of ψ µt Q is ˆ ψ µt ˆ Q . These twolifts differ by a constant m ∈ Z k since Qψ t = ψ µt Q for all t ∈ R :ˆ Q ˆ ψ ( t, x ) = ˆ ψ ( µt, ˆ Q ( x )) + m for all t ∈ R , x ∈ R k . Since ˆ ψ ( t, x ) = Bx + tBd + ˆ U ( x + td ) + ˆ c and ˆ ψ ( µt, ˆ Q ( x )) = Bx + ˆ U ( x ) + ˆ c + µtd, it follows that ˆ U ( x + td ) − ˆ U ( x ) = − t ( B − µI ) d + m for all t ∈ R , x ∈ R k , where I is the identity matrix. Evaluation of this at x = 0 givesˆ U ( td ) = − t ( B − µI ) d + m for all t ∈ R . However, ˆ U is bounded since it is continuous and periodic. This boundedness impliesthat ( B − µI ) d = 0, and so ˆ U ( td ) = m for all t ∈ R . Evaluation of this at t = 0 shows Spatzier’s Conjecture and Generalized Symmetries that m = 0 because ˆ U (0) = 0. Thus0 = ˆ U ( td ) = ˆ U ( ˆ ψ t (0)) for all t ∈ R . Since ˆ U is periodic and continuous on R n , it is a lift of a homeomorphism U on T n . Alift of U ψ t is ˆ U ˆ ψ t , and so0 = π ˆ U ( ˆ ψ t (0)) = U ( ψ t (0)) for all t ∈ R . The minimality of ψ implies that U is 0 on a dense subset of T n . By continuity, U = 0which implies that ˆ U = 0. Thus ˆ Q is C ∞ , and so Q is a diffeomorphism, whence Q ∈ S ψ with ρ ψ ( Q ) = µ . Since µ ∈ M φ , then M φ ⊂ M ψ .The multiplier µ ∈ M ψ is a real algebraic integer of degree at most k because itsatisfies ( B − µI ) d = 0 for nonzero d , i.e., the characteristic polynomial of B is a monicpolynomial of degree k with integer coefficients. The only rational roots this characteristicpolynomial can have are ± ψ is an irrational flow, i.e., M ψ ∩ Q = { , − } (seeCorollary 4.4 in [2]). However, if µ = ±
1, then the minimal polynomial for µ has degreebetween 2 and k inclusively. (cid:3) Theorem 4.2.
Suppose for an equilibrium-free flow Φ on P there is a Φ -invariantcompact submanifold N without boundary and R ∈ S Φ with | ρ Φ ( R ) | 6 = 1 such that R ( N ) ∩ N = ∅ . If dim( N ) = 2 and N is orientable, then ρ Φ ( R ) is a real algebraic integerof degree . If dim( N ) ≥ with N diffeomorphic to T dim( N ) and Φ | N is a minimal flow,then ρ Φ ( R ) is a real algebraic integer of degree between and dim( N ) inclusively.Proof. Let µ = ρ Φ ( R ) with | µ | 6 = 1 and k = dim( N ). By Theorem 2.1 there are noperiodic orbits for the equilibrium-free flow Φ | N . If k = 2 and N is orientable, thePoincar´e-Bendixson Theorem implies that N is diffeomorphic to T , and that Φ | N isminimal. If k ≥
3, it is assumed that N is diffeomorphic to T k and that Φ | N is minimal.The submanifold R ( N ) is Φ-invariant because N is Φ-invariant and R ∈ S Φ , i.e., for p ∈ R ( N ) and q = R − ( p ) ∈ N ,Φ( t, p ) = Φ( t, R ( q )) = R Φ( t/µ, q ) ⊂ R ( N ) for all t ∈ R . By hypothesis, there is ˜ p ∈ R ( N ) ∩ N . By the Φ-invariance of N and R ( N ) and theminimality of Φ | N it follows that O Φ (˜ p ) = N and O Φ (˜ p ) = R ( N ). This gives R ( N ) = N ,i.e., that N is R -invariant.The nontrivial generalized symmetry R restricts to a nontrivial generalized symmetryof Φ | N because N is Φ-invariant and R -invariant. If V = R | N and φ is the flow on N determined by X φ = X Φ | N , then R ∗ X Φ = µX Φ becomes T V X φ ( p ) = µX φ ( V ( p )) for p ∈ N. Since V ∈ Diff( N ), then V ∈ S φ with ρ φ ( V ) = µ .By [7], minimality of φ = Φ | N with N diffeomorphic to T n implies that φ is topologi-cally conjugate to an irrational flow ψ . Applying Lemma 4.1 shows that µ is an algebraicinteger of degree between 2 and k inclusively. (cid:3) Corollary 4.3.
Suppose Φ is a minimal flow on T n , and R ∈ S Φ . If | ρ Φ ( R ) | 6 = 1 , then ρ Φ ( R ) is an algebraic integer of degree between and n inclusively.Proof. Any nontrivial generalized symmetry R of Φ together with N = T n satisfy theconditions of Theorem 4.2. (cid:3) .F. Bakker 9 A Group-Theoretical Characterization of Irrational Flows
Minimality of an equilibrium-free flow on T n places a semidirect product structurethe generalized symmetry group of that flow. A group S is the semidirect product of twosubgroups N and H if N is a normal subgroup of S , if S = N H , and if
N ∩ H is theidentity element of S . Notational this is written S = N ⋊ Γ H , where Γ : H →
Aut( N ) is the conjugating homomorphism of the semidirect product,i.e., Γ( h )( n ) = hnh − for h ∈ H and n ∈ N . A normal subgroup of S ψ is ker ρ ψ for anyflow ψ . A normal subgroup of Diff( T n ) is the abelian group Trans( T n ) of translations.Each translation on T n is of the form T c ( θ ) = θ + c for c ∈ T n . If ψ is a flow on T n with X ψ a constant, then Trans( T n ) ⊂ ker ρ ψ because ( T c ) ∗ X ψ = X ψ for all c ∈ T n . Lemma 5.1.
Let ψ be a flow on T n for which X ψ is a constant. If ker ρ ψ = Trans( T n ) ,then ψ is irrational.Proof. Suppose that ψ is not irrational. Then the components of X ψ are not rationallyindependent. Up to a permutation of the coordinates θ , . . . , θ n on T n , i.e., a smooth con-jugacy, it can be assumed the first l components of X ψ are the smallest subset of the com-ponents that are linearly dependent over Q . Specifically, writing X ψ = [ a , a , . . . , a n ] T ,there is a least integer l with 1 ≤ l ≤ n such that k a + · · · + k l a l = 0with k i ∈ Z \ { } for all i = 1 , . . . , l . The existence of an R ∈ ker ρ ψ \ Trans( T n ) will beexhibited separately in the cases of l < n and l = n .Case 1 ≤ l ≤ n −
1. The R ∈ Diff( T n ) induced by the GL( n, Z ) matrix B = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . k k . . . k l − k l . . . satisfies R ∗ X ψ = X ψ . So R ∈ ker ρ ψ , but R Trans( T n ).Case l = n . Here none of the k , k , . . . , k n are zero, and the integer-entry matrix B = . . . . . . . . . k k . . . k n − k n induces a k n -to-1 smooth surjection g of T n to itself. The flow φ on T n generated by X φ = [ a , . . . , a n − , T is a factor of ψ i.e., gψ t = φ t g for all t ∈ R , because BX ψ = X φ .The components a , . . . , a n − of X φ are rationally independent. For, if they were not,there would then be integers k ′ , . . . , k ′ n − , not all zero, such that k ′ a + ··· + k ′ n − a n − = 0,which would contradict the minimality of l = n . With e , . . . , e n as the standard basis Spatzier’s Conjecture and Generalized Symmetries for R n and E = Span( e , . . . , e n − ) an ( n − R n , the closureof O φ (0) is π ( E ). Thus O φ (0) is an embedded submanifold of T n diffeomorphic to T n − .The semiconjugacy g between ψ and φ along with g (0) = 0 imply that O ψ (0) is anembedded submanifold of T n that is diffeomorphic to T n − . In particular, the ( n − U of R n determined by B ( U ) = E satisfies π ( U ) = O ψ (0), where U = Span( u , u , . . . , u n − ) for B ( u i ) = e i , i = 1 , . . . , n −
1. Furthermore, X ψ = a u + · · · + a n − u n − ∈ U because k a + · · · + k n a n = 0.For ǫ > V a nonempty subset of R n , and m ∈ Z n , define N ǫ ( V ) to be the set ofpoints in R n less than a distance of ǫ from V , and define V + m to be the translationof V by m . By the definition of E , if E + m = E , then N / ( E + m ) ∩ N / ( E ) = ∅ .Since B ( U ) = E and B ( Z n ) ⊂ Z n , there exists ǫ > U + m = U , then N ǫ ( U + m ) ∩ N ǫ ( U ) = ∅ .The vector e n U . Let x , . . . , x n be the coordinates on R n that correspond to thebasis u , . . . , u n − , e n . Let f : R → R be a smooth bump function with f (0) = 1 andwhose support has length smaller than ǫ . In terms of the coordinates x , . . . , x n , definea smooth vector field on N ǫ ( U ) by Y = f ( x n ) ∂∂x n . Extend this vector field to all of R n by translation to N ǫ ( U + m ) for those m ∈ Z m forwhich U + m = U , and to the remainder of R n as 0. The extended vector field is globallyLipschitz, and so determines a flow ξ on R n .Since the vector field generating ξ is invariant under translations by m ∈ Z , the time-one map ξ is also invariant under these translations. Thus ξ is a lift of an R ∈ Diff( T n ).In terms of the coordinates x , . . . , x n , the derivative of ξ at any point x ∈ R n is of theform T x ξ = . . . . . . . . . . . . ∗ , and so T x ξ ( u ) = u for u ∈ U . Since X ψ ∈ U , this means that T x ξ ( X ψ ) = X ψ . Since X ψ is a constant vector field, then ( ξ ) ∗ X ψ = X ψ . Since ξ is a lift of R , it now followsthat R ∗ X ψ = X ψ . Therefore R ∈ ker ρ ψ but R Trans( T n ). (cid:3) The group Aut( T n ) of automorphisms of T n is naturally identified with GL( n, Z ). For T c ∈ Trans( T n ) and B ∈ GL( n, Z ), the composition of T c with B is written T c B = B + c . Theorem 5.2.
Let ψ be a flow on T n with X ψ a nonzero constant vector. Then ψ isirrational if and only if there exists a subgroup H of GL( n, Z ) isomorphic to M ψ suchthat S ψ = Trans( T n ) ⋊ Γ H .Proof. Suppose that ψ is irrational. This implies (by Theorem 5.5 in [2]) that S ψ = ker ρ ψ ⋊ Γ H ψ , where H is a subgroup of GL( n, Z ) isomorphic to M ψ . Furthermore, the irrationality of ψ implies (by Corollary 4.7 in [2]) that ker ρ ψ = Trans( T n ). .F. Bakker 11 Now suppose that ψ is not irrational. By Lemma 5.1, there is R ∈ ker ρ ψ \ Trans( T n ). Ifthere were a subgroup H of GL( n, Z ) isomorphic to M ψ such that S ψ = Trans( T n ) ⋊ Γ H ,then R = T c B for some c ∈ T n and B ∈ H . Hence 1 = ρ ψ ( R ) = ρ ψ ( T c ) ρ ψ ( B ) = ρ ψ ( B ).However, since H is isomorphic to M ψ , there is only one element of H which correspondsto the multiplicative identity 1 of R × , and that is I , the identity matrix. This meansthat B = I , and so R = T c , a contradiction. (cid:3) Global Rigidity for Certain Z d Anosov Actions
Global rigidity is about when a Z d Anosov action, which as is well-known is topologi-cally conjugate to an algebraic Z d action (see [11]), is smoothly conjugate to an algebraic Z d action. A Z d action on T n is a monomorphism α : Z d → Diff( T n ). It is Anosov ifthere is m ∈ Z d \ { } with α ( m ) Anosov, is algebraic if α ( Z d ) ⊂ GL( n, Z ), and moregenerally is affine if α ( Z d ) ⊂ Trans( T n ) ⋊ Γ GL( n, Z ). Algebraic Z d -actions are found inalgebraic number theory (see [13] and [23]). Proof of Theorem 1.1.
Although well-known, the argument for the existence of a commonfixed point of a finite index subgroup of a Z d Anosov action is included for completeness.Let m ∈ Z d be such that α ( m ) is Anosov. Then α ( m ) is topologically conjugate toa hyperbolic automorphism of T n (see [19]), and so α ( m ) has a finite number of fixedpoints, f , . . . , f l . Let α ( m ) , . . . , α ( m d ) be a generating set for α ( Z d ). Since α ( Z d ) isabelian, then for all i = 1 , . . . , d and j = 1 , . . . , l , α ( m ) α ( m i )( f j ) = α ( m i ) α ( m )( f j ) = α ( m i )( f j ) . This means that α ( m i )( f j ) is one of the finitely many fixed points of α ( m ). Sinceeach α ( m i ) is invertible, there is a positive finite integer r i such that α ( m i ) r i ( f ) = f .Thus the finite index subgroup of α ( Z d ) generated by α ( m ) r , . . . , α ( m d ) r d has f as acommon fixed point.By hypothesis, there is a g ∈ Diff( T n ) and an irrational flow φ on T n for which X φ = g ∗ X Φ . By Theorem 5.2, there is a subgroup H φ of GL( n, Z ) isomorphic to M φ such that S φ = Trans( T n ) ⋊ Γ H φ . For h ∈ Diff( T n ) given by h = T − g ( f ) ◦ g it followsthat h ∗ X Φ = X φ because ( T c ) ∗ X φ = X φ for any c ∈ T n . Then Φ and φ are projectivelyconjugate, and so, as mentioned in Section 2, ∆ h ( S φ ) = S Φ . The inclusion α ( Z d ) ⊂ S Φ implies that ∆ h − ( α ( Z d )) ⊂ Trans( T n ) ⋊ Γ H φ . This means that α is C ∞ -conjugate to an affine Z d -action.For each α ( m ) in the finite index subgroup generated by α ( m ) r , . . . , α ( m d ) r d thereare B m ∈ H φ and c m ∈ T n such that ∆ h − ( α ( m )) = B m + c m . Since h ( f ) = 0, then c m = ( B m + c m )(0) = h ◦ α ( m ) ◦ h − (0) = h ◦ α ( m )( f ) = h ( f ) = 0 . This means that ∆ h − ( α ( m )) ∈ H φ . Hence the finite index subgroup of α ( Z d ) generatedby α ( m ) r , . . . , α ( m d ) r d , which is isomorphic to Z d , is C ∞ -conjugate to an algebraic Z d -action. Since H φ contains a Z d subgroup, and M φ is isomorphic to H φ , then M φ contains a Z d subgroup. By absolute invariance of the multiplier group under projectiveconjugacy, M Φ contains a Z d subgroup. (cid:3) Any quasiperiodic flow on T n whose generalized symmetry group contains a Z d Anosovaction must possess nontrivial generalized symmetries since its multiplier group containsa Z d subgroup by Theorem 1.1 (cf. Theorem 3.2). This puts a necessary condition on thequasiperiodic flows to which Theorem 1.1 does apply. The quasiperiodic flows of Koch Spatzier’s Conjecture and Generalized Symmetries type mentioned in the next section satisfy this necessary condition. However, there arequasiperiodic flows that do not, as is illustrated next.
Example 6.1. On T n , let ψ be the flow generated by X ψ = ∂∂θ + π ∂∂θ + · · · + π n − ∂∂θ n . This flow is quasiperiodic, since if its frequencies 1 , π, . . . , π n − were linearly dependentover Q then π would be algebraic. Quasiperiodicity of ψ implies that each µ ∈ M ψ is areal algebraic integer of degree at most n , and moreover, M ψ ∩ Q = { , − } (see Corollary4.4 in [2]). Suppose µ ∈ M ψ \ { , − } . Then there is R ∈ S ψ such that R ∗ X ψ = µX ψ .Quasiperiodicity of ψ implies that T R = B ∈ GL( n, Z ) (see Theorem 4.3 in [2]). Then R ∗ X ψ = µX ψ becomes BX ψ = µX ψ , and for B = ( b ij ), it follows that µ = b + b π + · · · + b n π n − . If b = · · · = b n = 0, then µ = b ∈ Z . But M ψ ∩ Q = { , − } , and so µ = ± b , . . . , b n is nonzero. Because each multiplier of ψ is a real algebraic integer of degree at most n , there is a monic polynomial l ( z ) in thepolynomial ring Z [ z ] such that l ( µ ) = 0. But this implies that π is a root of a polynomialin Z [ z ], making π algebraic. This shows that M ψ = { , − } , and so ψ does not possessnontrivial generalized symmetries.7. Classification of Certain Equilibrium-Free Flows
Quasiperiodic flows of Koch type are algebraic in nature and provide foliations whichare often preserved by a topologically irreducible Z d Anosov action (see [3], [4], and[13] for such examples). A flow on T n is quasiperiodic of Koch type if it is projectivelyconjugate to an irrational flow whose frequencies form a Q -basis for a real algebraicnumber field F of degree n over Q . For a quasiperiodic flow Φ of Koch type, the realalgebraic number field F of degree n associated to it is unique, and its multiplier groupis a finite index subgroup of the group of units o × F in the ring of integers of F (seeTheorem 3.3 in [3]). By Dirichlet’s Unit Theorem (see [24]), there is d ≥ o × F is isomorphic to Z ⊕ Z d , and so every quasiperiodic flow of Koch type always possessesnontrivial generalized symmetries.Topological irreducibility of a Z d action α is a condition on the topological factorsthat α has. A Z d action α ′ on T n ′ is a topological factor of α if there is a continuoussurjection h : T n → T n ′ such that h ◦ α = α ′ ◦ h . A topological factor α ′ of α is finite if the continuous surjection h is finite-to-one everywhere. A Z d action α is topologicallyirreducible if every topological factor α ′ of α is finite.For an algebraic Z d action α , there is a stronger sense of irreducibility, one that usesthe group structure of T n . An algebraic Z d action α ′ on T n ′ is an algebraic factor of α ifthere is a continuous homomorphism h : T n → T n ′ such that h ◦ α = α ′ ◦ h . An algebraicfactor α ′ of α is finite if the continuous homomorphism h is finite-to-one everywhere. Analgebraic Z d action α is algebraically irreducible if every algebraic factor α ′ of α is finite.Algebraic irreducibility of a higher rank algebraic Z d action α is equivalent to there beingan m ∈ Z d such that α ( m ) has an irreducible characteristic polynomial (see Proposition3.1 on p. 726 in [13]; cf. [8]). Proof of Theorem 1.2.
Identify T T n with T n × R n , and place on the fiber the standardEuclidean norm k · k . By the hypotheses, there is h ∈ Diff( T n ) and a hyperbolic B ∈ .F. Bakker 13 GL( n, Z ) such that ∆ h − ( α ( m )) = B . Every point of T n is Lyapunov regular for B .The Oseledets decomposition associated with χ B is T θ T n = k M i =1 E iB , where E iB , i = 1 , . . . , k , are the invariant subspaces of B which are independent of θ .Since ∆ h − ( α ( m )) = B , every point of T n is Lyapunov regular for α ( m ). Set E iα ( m ) ( θ ) = T h ( θ ) h − (cid:0) E iB ) , θ ∈ T n . The Oseledets decomposition associated with χ α ( m ) is then T θ T n = k M i =1 E iα ( m ) ( θ ) . For µ = ρ Φ ( α ( m )), the hypothesis that the multiplicity of the value log | µ | of χ α ( m ) is one at a point ˆ θ ∈ T n implies that there is 1 ≤ l ≤ k such that dim (cid:0) E lα ( m ) (ˆ θ ) (cid:1) = 1and χ α ( m ) (ˆ θ, v ) = log | µ | for v ∈ E lα ( m ) (ˆ θ ) \ { } . By the definition E lα ( m ) (ˆ θ ) = T h (ˆ θ ) h − (cid:0) E lB ), it follows that dim (cid:0) E lB (cid:1) = 1. Furthermore,since ∆ h − ( α ( m )) = B and χ B is independent of θ , it follows that χ B ( θ, v ) = log | µ | for all v ∈ E lB \ { } and for all θ ∈ T n . The definition E lα ( m ) ( θ ) = T h ( θ ) h − (cid:0) E lB )and the independence of E lB from θ imply for all θ ∈ T n that dim (cid:0) E lα ( m ) ( θ ) (cid:1) = 1 and χ α ( m ) ( θ, v ) = log | µ | for all v ∈ E lα ( m ) ( θ ) \ { } . Hence, the multiplicity of log | µ | for χ α ( m ) is one for all θ ∈ T n .By Theorem 3.1, the α ( m )-invariant one-dimensional distribution E given by E ( θ ) =Span( X Φ ( θ )) satisfies χ α ( m ) ( θ, X Φ ( θ )) = log | µ | for all θ ∈ T n . If E ( θ ) = E lα ( m ) ( θ ) atsome θ ∈ T n , then E ( θ ) + E lα ( m ) ( θ ) is a two-dimensional subspace of T θ T n for which χ α ( m ) ( θ, v ) = log | µ | for all v ∈ (cid:0) E ( θ )+ E lα ( m ) ( θ ) (cid:1) \{ } . This contradicts the multiplicityof log | µ | for χ α ( m ) being one at every θ . Thus E lα ( m ) ( θ ) = E ( θ ) = Span( X Φ ( θ )) for all θ ∈ T n .The vector field h ∗ X Φ satisfies h ∗ X Φ ( θ ) ∈ E lB for all θ ∈ T n because Span( X Φ ( θ )) = T h ( θ ) h − ( E lB ) for all θ ∈ T n . Let ψ be the flow for which X ψ = h ∗ X Φ . Since E lB is aone-dimensional invariant subspace of B and χ B ( θ, v ) = log | µ | for all v ∈ E lB \ { } , itfollows for all θ ∈ T n that k B k X ψ ( θ ) k = | µ | k k X ψ ( θ ) k for all k ∈ Z . Hyperbolicity of B implies that there is ¯ θ ∈ T n such that O B (¯ θ ) = { B k (¯ θ ) : k ∈ Z } is dense in T n . Since α ( m ) ∗ X Φ = µX Φ and X ψ = h ∗ X Φ , the matrix B satisfies BX ψ = µX ψ B . Then B k X ψ = µ k X ψ B k , and so X ψ B k = µ − k B k X ψ . Thus, k X ψ ( B k ¯ θ ) k = | µ | − k k B k X ψ (¯ θ ) k = | µ | − k | µ | k k X ψ (¯ θ ) k = k X ψ (¯ θ ) k for all k ∈ Z . Denseness of O B (¯ θ ) and continuity of X ψ imply that k X ψ ( θ ) k = k X ψ (¯ θ ) k for all θ ∈ T n .The one-dimensionality of E lB to which X ψ belongs implies that X ψ is a constant vector.Thus X ψ is an eigenvector of B .The assumed topological irreducibility of α and the inclusion ∆ h − ( α ( Z d )) ⊂ GL( n, Z )imply that ∆ h − ( α ( Z d )) is algebraically irreducible. Thus there is B ′ ∈ ∆ h − ( α ( Z d )) withan irreducible characteristic polynomial. Since BB ′ = B ′ B and B ′ has an irreducible Spatzier’s Conjecture and Generalized Symmetries characteristic polynomial, the eigenvector X ψ of B is an eigenvector of B ′ too. Thenthere is ϑ ∈ R × such that the components of ϑ − X ψ form a Q -basis for an algebraicnumber field of degree n over Q (see Propositions 1 and 8 in [25]). Thus the flow φ determined by X φ = ϑ − X ψ is irrational of Koch type for which h ∗ X Φ = X ψ = ϑX φ . (cid:3) References [1] R.L. Adler and R. Palais,
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