A mixing-like property and inexistence of invariant foliations for minimal diffeomorphisms of the 2-torus
aa r X i v : . [ m a t h . D S ] F e b A MIXING-LIKE PROPERTY AND INEXISTENCE OFINVARIANT FOLIATIONS FOR MINIMAL DIFFEOMORPHISMSOF THE 2-TORUS
ALEJANDRO KOCSARD AND ANDR´ES KOROPECKI
Abstract.
We consider diffeomorphisms in O ∞ ( T ), the C ∞ -closure of theconjugancy class of translations of T . By a theorem of Fathi and Herman,a generic diffeomorphism in that space is minimal and uniquely ergodic. Wedefine a new mixing-type property, which takes into account the “directions”of mixing, and we prove that generic elements of O ∞ ( T ) satisfy this property.As a consequence, we obtain a residual set of strictly ergodic diffeomorphismswithout invariant foliations of any kind. We also obtain an analytic version ofthese results. Introduction
In [FH77], Fathi and Herman combined generic arguments with the so-called fastapproximation by conjugations method of Anosov and Katok [AK70], to study aparticular class of diffeomorphisms of a compact manifold: the C ∞ -closure of theset of diffeomorphisms which are C ∞ conjugate to elements of a locally free T -action on the manifold. They proved that a generic element of that space is minimaland uniquely ergodic (i.e. there is a residual subset of such diffeomorphisms), inparticular proving that every compact manifold admitting a locally free T -actionsupports a minimal and uniquely ergodic diffeomorphism.Surprisingly, the space studied by Fathi and Herman contains many elementswith unexpected dynamical properties; for example, a generic diffeomorphism inthat space is weak mixing [Her92, FS05], and if the underlying space is T n , theaction of its derivative on the unit tangent bundle is minimal [Kor07]. For a verycomplete survey on the technique of Anosov-Katok and its variations, see [FK04]. In[FS05], Fayad and Saprikyna use a real analytic version of this method to constructminimal weak mixing diffeomorphisms.In this short article, we restrict our attention to diffeomorphisms of T . In thissetting, the closure of maps C ∞ conjugated to elements of any locally free T -actioncoincides with the C ∞ -closure of the conjugancy class of the rigid translations R ( λ ,λ ) : ( x, y ) ( x + λ , y + λ ) , that is, the set O ∞ ( T ), where O ( T ) = (cid:8) hR α h − : h ∈ Diff ∞ ( T ) , α ∈ T (cid:9) . As we mentioned above, a generic element of O ∞ ( T ) is topologically weakmixing; however, no topologically mixing elements are known. It is also unknown ifa minimal diffeomorphism of T can be topologically mixing. In fact, no examples The authors were supported by CNPq-Brasil. of minimal C ∞ diffeomorphisms of T in the homotopy class of the identity areknown other than the ones in O ∞ ( T ).Recall that a homeomorphism f : T → T is topologically weak mixing if f × f is transitive. An equivalent definition is the following: for each open U ⊂ T and ǫ >
0, there is n > f n ( U ) is ǫ -dense in T . We will be interested in asimilar property, which implies weak-mixing but is stronger in that it requires opensets to be mixed in every homological direction. Definition 1.1.
A homeomorphism f : T → T is weak spreading if for a liftˆ f : R → R of f the following holds: for each open set U ∈ R , ǫ > N > n > f n ( U ) is ǫ -dense in a ball of radius N .Let O ∞ µ ( T ) denote the area-preserving elements of O ∞ ( T ). Now we can stateour main theorem. Theorem 1.2.
Weak spreading diffeomorphisms are generic in O ∞ ( T ) and O ∞ µ ( T ) . As a consequence, we prove a result about invariant foliations announced byHerman in [FH77] without proof. By a topological foliation we mean a codimension-1 foliation of class C ; that is, a partition F of T into one-dimensional topologicalsub-manifolds which is locally homeomorphic to the partition of the unit square byhorizontal segments. We say that the foliation is invariant by f if f ( F ) ∈ F forevery F ∈ F . We then have: Corollary 1.3.
The set of diffeomorphisms in O ∞ ( T ) (resp. O ∞ µ ( T ) ) withoutany invariant topological foliation is residual in O ∞ ( T ) (resp. O ∞ µ ( T ) ). Since the set of minimal and uniquely ergodic diffeomorphisms in O ∞ ( T ) (or O ∞ µ ( T )) is also residual, this provides a residual set of minimal, uniquely ergodicdiffeomorphisms with no invariant foliations.Using the ideas of [FS05], it is possible to construct real analytic examples,working with diffeomorphisms which have an analytic extension to a band of fixedwidth in C (see the precise definitions in § Theorem 1.4.
The set of real analytic diffeomorphisms of T which are weakspreading is residual in O ωρ ( T ) .Remark . We use the word “weak” in the definition of weak spreading becausethere is an analogy with the topological weak mixing property. We could alsodefine strong spreading (or just spreading ) as the property that for any open set U ⊂ R , ǫ > N > n such that that ˆ f n ( U ) is ǫ -dense in aball of radius N whenever n > n . This would be in analogy with the definition oftopological mixing, but it is clearly a stronger property. In fact, the typical examplesof topologically mixing systems in T mix only in one direction (e.g. Anosov systemsand time-one maps of some minimal flows [Fay02]). It is not obvious that strongspreading diffeomorphisms exist; however, as P. Boyland kindly explained to us,an example of a strong spreading diffeomorphism can be constructed using Markovpartitions and the techniques of [Boy08].1.1. Acknowledgments.
We are grateful to E. Pujals and P. Boyland for usefuldiscussions, and the anonymous referee for bringing the results of [FS05] to ourattention and suggesting various improvements, in particular the content of § MIXING-LIKE PROPERTY FOR MINIMAL DIFFEOMORPHISMS OF THE 2-TORUS 3 The method of Fathi-Herman
As usual, we identify T ≃ R / Z with quotient projection π : R → T , anddenote by Diff ∞ ( T ) the space of C ∞ diffeomorphisms of T . A lift of one such dif-feomorphism f to R is a map ˆ f : R → R such that f π = π ˆ f . If f is homotopic tothe identity, this is equivalent to saying that ˆ f commutes with integer translations,i.e. ˆ f ( z + v ) = ˆ f ( z ) + v for v ∈ Z . Two different lifts of a diffeomorphism of T always differ by a constant v ∈ Z . We will denote by ˆ R α the translation z z + α of R and by R α the rotation of T lifted by ˆ R α .The method used in [FH77], adapted to our case, can be resumed as follows: Lemma 2.1.
Let P ⊂ O ∞ ( T ) (or O ∞ µ ( T ) ) be such that (1) P = T n ≥ P n , where the P n are open; (2) For each g ∈ Diff ∞ ( T ) (resp. Diff ∞ µ ( T ) ) and m ∈ N , there is N > suchthat { gf g − : f ∈ P n } ⊂ P m whenever n > N ; (3) For each n ∈ N , p/q ∈ Q , there exists h ∈ Diff ∞ ( T ) (resp. Diff ∞ µ ( T ) )such that • hR (1 /q, = R (1 /q, h ; • hR α k h − ∈ P n for some sequence α k → ( p/q, .Then, P is residual in O ∞ ( T ) (resp. O ∞ µ ( T ) ).Proof. Given m ∈ N , p/q ∈ Q , and g ∈ Diff ∞ ( T ), let n be as in (2), and then h and { α k } as in (3). Then P n ∋ hR α k h − C ∞ −−−−→ k →∞ hR ( p/q, h − = R ( p/q, , so that P m ∋ g ( hR α k h − ) g − C ∞ −−−−→ k →∞ gR ( p/q, g − . This proves that gR ( p/q, g − ∈ P ∞ m . Since this holds for all g and p/q , it followsthat P m is dense in O ∞ ( T ) because so is the set (cid:8) hR ( p/q, h − : h ∈ Diff ∞ ( T ) , p/q ∈ Q (cid:9) . Since this holds for all m and each P m is open, this proves that P is residual in O ∞ ( T ). The proof in the area-preserving case is the same. (cid:3) The property of having no invariant topological foliations is hard to deal with inthe C ∞ topology in order to apply the above lemma. However, the weak spread-ing property can be adequately described as an intersection of countably manyproperties that fit well into the lemma; thus we first prove Theorem 1.2 using theabove method, and then we prove that weak spreading is not compatible with theexistence of invariant foliations of any kind, which implies Corollary 1.3.3. Proof of Theorem 1.2
Let P n denote the set of all f ∈ O ∞ ( T ) such that if ˆ f is a lift of f , for anyball B of radius 1 /n in R , there is k > f k ( B ) is 1 /n -dense in a ball ofradius n . Note that if this property holds for some lift, it holds for any lift of f .It is clear that P = ∩ P n is the set of weak spreading elements of O ∞ ( T ). Denoteby B ( z, ǫ ) the ball of radius ǫ centered at z . Given a lift ˆ f of some f ∈ P n , and z ∈ R , let k z be the smallest positive integer such that ˆ f k z ( B ( z, /n )) is 1 /n -dense A. KOCSARD AND A. KOROPECKI ˆ v (ˆ u ( J δ ))ˆ u ( J δ ) J δ Figure 1.
Image of J δ by ˆ h in a ball of radius n . By continuity of ˆ f , the map z k z is upper semi-continuous,and therefore it attains a maximum K when z ∈ [0 , . But since ˆ f lifts a maphomotopic to the identity, k z = k z + v when v ∈ Z , so that k z ≤ K for all z ∈ R .Hence, if g is close enough to f in the C topology and ˆ g is the lift of g closest toˆ f , it also holds that ˆ g k z ( B ( z, /n )) is dense in a ball of radius n for any z ∈ R .Hence P n is open in the C topology (and, in particular, in the C ∞ topology).To see that condition (2) of Lemma 2.1 holds, note that any lift ˆ g of a diffeo-morphism g of T is bi-Lipschitz. Fix m ∈ N , let C be a Lipschitz constant for ˆ g and ˆ g − , and let n > C < n/m . If ˆ f is a lift of f ∈ P n , and if U is an open set, then there is k such that ˆ f k (ˆ g − ( U )) is 1 /n -dense in a ball ofradius n . Thus, ˆ g ˆ f k ˆ g − ( U ) is C/n -dense in a ball of radius n/C , which impliesthat gf g − ∈ P m as required.To finish the proof, it remains to see that condition (3) of Lemma 2.1 holds. Todo this, it suffices to construct, for each q, n ∈ N , a diffeomorphism h ∈ Diff ∞ ( T )which commutes with R (1 /q, and such that hR ( α, h − ∈ P n whenever α is irra-tional. Note that it is enough to prove this for some multiple of q instead of q . Wewill assume that q is a multiple of n and q ≥ n , since otherwise we may use 2 qn instead of q . We define h by constructing a lift ˆ h = ˆ v ◦ ˆ u , where ˆ v, ˆ u : R → R arethe maps ˆ u ( x, y ) = ( x, y + m cos(2 πqx )) , ˆ v ( x, y ) = ( x + n cos(2 πqy ) , y )and m is a sufficiently large integer that we will choose later. It is clear that ˆ u and ˆ v are lifts of C ∞ torus diffeomorphisms in the homotopy class of the identity,because they commute with integer translations. They also commute with ˆ R (1 /q, and ˆ R (0 , /q ) as well. The same properties hold for ˆ h = ˆ v ◦ ˆ u . Moreover, since bothˆ u and ˆ v are area-preserving, so is ˆ h and the rest of the proof also works in thearea-preserving setting.Let δ = 2 n ( πqm ) − , I δ = [ − δ, δ ] ×{ } , and J δ = [(4 q ) − − δ/ , (4 q ) − + δ/ ×{ } . Claim 1. If m is large enough, then ˆ h ( I δ ) is contained in the ball of radius / (2 n ) centered at ( n, m ) , and ˆ h ( J δ ) is /n -dense in [ − n, n ] × [ − n, n ] . MIXING-LIKE PROPERTY FOR MINIMAL DIFFEOMORPHISMS OF THE 2-TORUS 5
Proof.
First observe that from the inequality1 − cos( x ) ≤ x / ∀ x ∈ R it follows that (denoting by ( x , x ) i the coordinate x i ) | (ˆ u ( x, − ˆ u (0 , | ≤ m ( πqx ) < m ( πqδ ) = 8 n /m if | x | < δ . Since ˆ u (0 ,
0) = (0 , m ), this means that ˆ u ( I δ ) is contained in the rectangle[ − δ, δ ] × [ m − b, m + b ] where b = 8 n /m . By the definition of ˆ v and a similarargument (since m is an integer), we can conclude that ˆ v (ˆ u ( I δ )) ⊂ [ n − a, n + a ] × [ m − b, m + b ], where a = δ + 2 n ( πqb ) = 2 n ( πqm ) − + 128 π q n m − . Since both a and b can be made arbitrarily small if m is large enough, ˆ h ( I δ ) iscontained in a ball around ( n, m ) of radius 1 / (2 n ) if m is large enough.For the second part of the claim, note thatcos( x + π/
2) = sin( x ) ≥ x/ ≤ x ≤ π/ (cid:0) ˆ u ((4 q ) − + δ/ , − ˆ u ((4 q ) − , (cid:1) = m cos( πqδ + π/ ≥ mπqδ/ n, and similarly (cid:0) ˆ u ((4 q ) − − δ/ , − ˆ u ((4 q ) − , (cid:1) = − n. Thus ˆ u ( J δ ) is an arc that transverses vertically the rectangle [ − δ/ , δ/ × [ − n, n ].Let L = { } × [ − n, n ]. Note that ˆ v ( L ) is 1 /q -dense in [ − n, n ] × [ − n, n ], sinceevery rectangle of the form [ − n, n ] × [ − n + k/q, − n + ( k + 1) /q ], 0 ≤ k ≤ qn − v ( L ). By the previous paragraph, ˆ u ( J δ ) contains a pointof the form ( s, y ) with | s | < δ/ , y ) ∈ L . Since ˆ v ( s, y ) = ˆ v (0 , y )+( s, m is so large that δ/ < /q , ˆ h ( J δ ) = ˆ v (ˆ u ( J δ ))is 2 /q -dense in [ − n, n ] × [ − n, n ] (see Figure 1). Since we assumed earlier that q ≥ n ,we conclude that h ( J δ ) is 1 /n -dense in [ − n, n ] × [ − n, n ] as claimed. This provesthe claim. (cid:3) Let B ⊂ R be a ball of radius 1 /n . Then B contains a ball B ′ of radius 1 / (2 n )around some point of coordinates ( i/q, j/q ), with i, j integers (because q ≥ n ).Since ˆ h commutes with R (1 /q, , and using Claim 1, we see thatˆ h ( I δ + ( i/q, j/q ) − ( n, m )) = ˆ h ( I δ ) − ( n, m ) + ( i/q, j/q ) ⊂ B ′ In particular, I δ +( i/q, j/q ) − ( n, m ) ⊂ ˆ h − ( B ). Since J δ lies on the same horizontalline as I δ and is shorter than I δ , if α is an irrational number we can find k ∈ N and r ∈ Z such that J δ + ( r, ⊂ ˆ R k ( α, ( I δ ), and we have J δ + ( i/q, j/q ) − ( n, m ) + ( r, ⊂ ˆ R k ( α, ( I δ + ( i/q, j/q ) − ( n, m )) . Thus, if ˆ f = ˆ h ˆ R ( α, ˆ h − ,ˆ f k ( B ) = ˆ h ˆ R k ( α, ˆ h − ( B ) ⊃ ˆ h ˆ R k ( α, ( I δ + ( i/q, j/q ) − ( n, m )) ⊃ ˆ h ( J δ + ( i/q, j/q ) − ( n + r, m )) = ˆ h ( J δ ) + ( i/q, j/q ) − ( n + r, m )which is just a translation of ˆ h ( J δ ), and thus by Claim 1 it is 1 /n -dense in someball of radius n . That is, ˆ f k ( B ) is 1 /n -dense in some ball of radius n , which means A. KOCSARD AND A. KOROPECKI that hR ( α, h − ∈ P n . Since α was an arbitrary irrational number, this completesthe proof. (cid:3) Invariant foliations
Corollary 1.3 is a direct consequence of Theorem 1.2 and the next two proposi-tions.
Proposition 4.1. If F is a foliation of T and ˆ F is the lift of F to R , then thereis a leaf F ∈ ˆ F which is contained in a strip bounded by two parallel straight lines L and L ′ , such that both lines belong to different connected components of R − F .Proof. If F has a compact leaf, there is z ∈ R and a leaf F of ˆ F such that F + ( p, q ) = F , for some pair of integers ( p, q ) = (0 , p = 0, if L is a line of slope q/p , it holds that s = sup { d ( z, L ) : z ∈ F } < ∞ , and theproposition follows by choosing L and L ′ a distance greater than s apart from L ,one on each side. If p = 0, then q = 0 an analogous argument holds.Now suppose F has no compact leaves. By [HH83, Theorem 4.3.3], F is equiva-lent to a foliation F ′ obtained by suspension of the trivial foliation R × T over anorientation preserving circle homeomorphism f : T → T with irrational rotationnumber. Such a foliation has a lift ˆ F ′ to R such that the intersection of the leafthrough (0 , y ) with the line { n }× R is at ( n, ˆ f n ( y )), where ˆ f : R → R is a lift of f . If φ ( y ) denotes the length of the arc of leaf joining (0 , y ) to (1 , ˆ f ( y )), then φ : R → R is a continuous function and it is Z -periodic, because ˆ F ′ is a lift of a foliation of T . Thus there is a constant C such that φ ( x ) < C for all x ∈ R . Note that thelength of the arc joining ( n, y ) to ( n + 1 , ˆ f ( y )) is also bounded by C .If ρ is the rotation number of ˆ f , by classic results for circle homeomorphisms(see, for example, [dMvS93]) we have | ˆ f n ( y ) − y − nρ | ≤ n ∈ Z and y ∈ R .Let F ′ be a leaf of ˆ F ′ containing the point (0 , y ). Then F ′ = ∪ n ∈ Z F ′ n where F n isthe arc joining ( n, ˆ f n ( y )) to ( n +1 , ˆ f n +1 ( y )). Note that the distance from ( n, ˆ f n ( y ))to the line L of slope ρ through (0 , y ) is at most 1, and the length of F ′ n is at most C . Thus the distance from any point of F ′ to L is at most C + 1.We know that F is equivalent to F ′ , which means there is a homeomorphism h : T → T mapping leaves of F ′ to leaves of F . If ˆ h : R → R is a lift of h ,then we can write ˆ h ( z ) = A ( z ) + ψ ( z ) where A ∈ GL(2 , Z ) and ψ is a Z -periodicfunction, bounded by some constant K . If L = AL , z = ˆ h ( z ′ ) is a point in F = h ( F ′ ), and w = A ( w ′ ) is a point in L then | z − w | = | A ( z ′ − w ′ ) + ψ ( z ) | ≤ k A k · | z ′ − w ′ | + K ≤ k A k ( C + 1) + K, the last inequality following from the fact that z ′ ∈ F ′ and w ′ ∈ L . It follows that s = sup z ∈ F d ( z, L ) < ∞ , and as before we complete the proof choosing L and L ′ parallel to L and a distance at least s apart from L , one on each side. (cid:3) Proposition 4.2. If f is weak spreading and homotopic to the identity, then f hasno invariant topological foliations.Proof. By Proposition 4.1, if F is a foliation invariant by f and ˆ F is the lift of thisfoliation to R (hence invariant by ˆ f ), there is a leaf ˆ F ∈ ˆ F which is contained in astrip bounded by two parallel lines L and L ′ , and which contains each of those linesin a different component of its complement. Let u be a unit vector orthogonal to L .We will assume without loss of generality that u has a nonzero second coordinate. MIXING-LIKE PROPERTY FOR MINIMAL DIFFEOMORPHISMS OF THE 2-TORUS 7 If S is the strip bounded by ˆ F and ˆ F + (0 , φ u : R → R theorthogonal projection onto the direction of u , it is clear thatwidth u ( S ) . = diam( φ u ( S )) < ∞ . Moreover, ∪ n ∈ Z S + (0 , n ) = R .For each n ∈ Z , since ˆ f n ( ˆ F ) cannot cross ˆ F + (0 , k ) for any k ∈ Z , we see thatˆ f n ( ˆ F ) ⊂ S + (0 , m ) for some m ∈ Z . This implies that width u ( f n ( ˆ F )) ≤ M =width u ( S ) . But then ˆ f n ( ˆ F + (0 , ⊂ S + (0 , m + 1) , so that ˆ f n ( S ) is contained in the strip bounded by ˆ F + (0 , m ) and ˆ F + (0 , m +2). This means that width u ( ˆ f n ( S )) ≤ M . However, if f is weak spreading,then there is n > f n ( S ) is 1 / M , sothat width u ( ˆ f n ( S )) > M , contradicting the previous claim. This completes theproof. (cid:3) The real analytic case
In this section we briefly explain how to obtain minimal weak spreading analyticdiffeomorphisms of T . We kindly thank the anonymous referee for bringing thisto our attention.First we introduce some notation, following [FS05]. Fix ρ >
0, and let g : R → R be any real analytic Z -periodic function which can be holomorphically extendedto A ρ = { ( z, w ) ∈ C : | Im z | < ρ, | Im w | < ρ } . We define k g k ρ = sup A ρ | g ( z, w ) | ,and we denote by C ωρ ( T ) the space of all functions of this kind which satisfy k g k ρ < ∞ .Let Diff ωρ ( T ) be the space of all diffeomorphisms f of T which are homotopicto the identity, and which have a lift whose periodic part is in C ωρ ( T ). There is ametric in Diff ωρ ( T ) defined by d ρ ( h, k ) = inf ( p,q ) ∈ Z (cid:13)(cid:13)(cid:13) ˆ h − ˆ k + ( p, q ) (cid:13)(cid:13)(cid:13) ρ , where ˆ h and ˆ k are lifts of h and k , respectively. Since C ωρ ( T ) is a Banach space, itis easy to see that the metric d ρ turns Diff ωρ ( T ) into a complete metric space.To apply the arguments of the previous sections we work in the space O ωρ ( T )defined as the closure in the d ρ metric of the set of diffeomorphisms of the form hR α h − where α ∈ T , and h ∈ Diff ωρ ( T ) is any diffeomorphism whose lifts to R have a bi-holomorphic extension to C .We observe that the proof of Lemma 2.1 applies to this setting if we use O ωρ ( T )instead of O ∞ ( T ) (and the topology induced by d ρ instead of the C ∞ topology).To complete the proof of Theorem 1.4, we note that everything in § h constructed to obtain property(3) of Lemma 2.1 has an analytic extension to all of C which is a bi-holomorphism. References [AK70] D. Anosov and A. Katok,
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Universidade Federal Fluminense, Instituto de Matem´atica, Rua M´ario Santos BragaS/N, 24020-140 Niteroi, RJ, Brasil
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