A chain transitive accessible partially hyperbolic diffeomorphism
aa r X i v : . [ m a t h . D S ] D ec A chain transitive accessible partially hyperbolic diffeomorphism
SHAOBO GAN and YI SHI
Abstract
In this paper, we construct a partially hyperbolic skew-product diffeomorphism f on T ,such that f is accessible and chain transitive, but not transitive. Let M be a closed Riemannian manifold, and f : M → M be a diffeomorphism. We say f istransitive, if for any two open sets U, V ⊂ M , there exists n >
0, such that f n ( U ) ∩ V = ∅ .Transitivity is a notion to describe the mixing property of the dynamics generated by f . Thetransitivity of f is equivalent to there exists a point x whose positive orbit { f n ( x ) : n > } isdense in M .We call a point x ∈ M is a non-wandering point of f , if for any neighborhood U x of x ,there exists n >
0, such that f n ( U x ) ∩ U x = ∅ . The non-wandering set Ω( f ) is the set of allnon-wandering points of f . It is clear that a point is a non-wandering point, then its orbit hassomekind recurrent property.For two points x, y ∈ M , we say y is chain attainable from x , if for any ǫ >
0, there exists afinite sequence { x i } ni =0 with x = x and x n = y , such that d ( f ( x i ) , x i +1 ) < ǫ for any 0 ≤ i ≤ n − x ∈ M is called a chain recurrent point, if it is chain attainable from itself. The setof chain recurrent points is called chain recurrent set of f , denoted by CR( f ). If every point ischain recurrent, we say f is chain transitive.It is clear that if a point is non-wandering, then it must be chain recurrent. Similarly, if f istransitive, then it must be chain transitive. However, from the powerful chain connecting lemma[3], there exists a residual set R ⊂
Diff ( M ), such that for any f ∈ R , we have Ω( f ) = CR( f ).Moreover, for the classical Anosov diffeomorphisms, we must have their non-wandering sets areequal to chain recurrent sets.A diffeomorphism f : M → M is partially hyperbolic, if the tangent bundle T M splits intothree nontrivial Df -invariant bundles T M = E ss ⊕ E c ⊕ E uu , such that Df | E ss is uniformlycontracting, Df | E uu is uniformly expanding, and Df | E c lies between them: k Df | E ss ( x ) k < k Df − | E c ( f ( x )) k − , k Df | E c ( x ) k < k Df − | E uu ( f ( x )) k − , for all x ∈ M. It is known that there are unique f -invariant foliations W ss and W uu tangent to E s and E u respectively.An important geometric property of partially hyperbolic diffeomorphisms is accessibility. Apartially hyperbolic diffeomorphism f is accessible, if for any two pints x, y ∈ M , they canbe joined by an arc consisting of finitely many segments contained in the leaves of foliations W ss and W uu . Accessibility plays a key role for proving the ergodicity of partially hyperbolicdiffeomorphisms, see [7, 11]. Moreover, it has been observed that most of partially hyperbolicdiffeomorphisms are accessible [6, 8, 11]. 1. Brin [5] has proved that for a partially hyperbolic diffeomorphism f : M → M , if f is accessible and Ω( f ) = M , then f is transitive. See also [1]. So it is natural to ask thefollowing question: if a partially hyperbolic diffeomorphism f is accessible and CR( f ) = M , is f transitive? In this paper, we construct an example which gives a negative answer to this question.This implies Brin’s result could not be generalized for the case where CR( f ) = M .Let A : T → T be a hyperbolic automorphism over T . We say f : T → T is a partiallyhyperbolic skew-product over A , if for every ( x, t ) ∈ T = T × S , we have f ( x, t ) = ( Ax, ϕ x ( t )) , and k A − k − < k ϕ ′ x ( t ) k < k A k . We will consider S = R / Z , and usually use the coordinate S = [ − , / {− , } .Our main result is the following theorem. Theorem 1.
There exists a partially hyperbolic skew-product C ∞ -diffeomorphism f : T → T ,such that f is accessible and chain transitive, but not transitive.Remark . We want to point out that for C -generic diffeomorphisms, chain transitivity impliestransitivity. Our construction need the help of nonhyperbolic periodic points. So we don’t knowfor C r -generic or C r -open dense accessible partially hyperbolic diffeomorphisms, whether chaintransitivity implies transitivity. The idea of our example is first we construct a chain transitive partially hyperbolic skew-productdiffeomorphism on T , such that its non-wandering set is not the whole T and not transitive.Then we make a small perturbation to achieve the accessibility, and still preserving the dynamicalproperties.First we need a diffeomorphism on S that is chain transitive but the non-wandering set isnot the whole circle.Let θ : S → S be defined as θ ( t ) = − cos(2 πt ) + 1 , t ∈ R / Z . It is a C ∞ -smooth function on S . We can see that θ ≥ S , and has two zero points 0and − { θ ( t ) · ∂∂t } is a smooth vector field on S , and its time- r map for0 < r ≪ r mapof θ ( t ) · ∂∂t is chain transitive, and the non-wandering set consists of only two fixed points 0 and − T = T × S . Lemma 2.1.
The vector field X defined by X ( x, t ) = θ ( t ) · ∂∂t , ∀ ( x, t ) ∈ T = T × S , is a smooth vector field on T . Moveover, for every r > , the time- r map X r of the flowgenerated by X satisfies the following properties: • X r ( x, t ) = ( x, ϕ x ( t )) for every ( x, t ) ∈ T . • For i = 0 , , X r ( x, i ) = ( x, i ) for every x ∈ T . Fix < δ ≪ , then for every ( x, t ) ∈ T × ([ − δ, ∪ [1 − δ, , we have X r ( x, t ) =( x, ϕ x ( t )) satisfies ϕ x ( t ) > t . In particularly, if we choose r small enough, there exists < τ = τ ( r, δ ) < δ/ , such that ϕ x ( t ) > t + τ, ∀ ( x, t ) ∈ T × {− δ, − δ } . − θ ( t ) · ∂/∂t T T T f r : T → T Figure 1: Chain transitive systems with nonempty wandering sets.For r > f r = X r ◦ ( A × id) : T → T .Then with the same constants δ = δ ( r ) and τ = τ ( r ) in the last lemma, f r satisfies the followingproperties(Figuer 1): • f r is a partially hyperbolic skew-product diffeomorphism on T , f r ( x, t ) = ( Ax, ϕ Ax ( t )),where ϕ x ( t ) is exactly the same as X r . Let the partially hyperbolic splitting be: T T = E ss ⊕ E c ⊕ E uu , and denote by W ss/uu the stable/unstable manifolds generated by E ss/uu . • In the fixed center fiber S p , f r | S p is chain transitive and has two fix points P i = ( p, i ) ∈ T × S for i = 0 , • For i = 0 , f r preserves T i = T × { i } invariant, and f r | T i = A | T i . Moreover, T i = W ss ( P i , f r ) = W uu ( P i , f r ) . • For every ( x, t ) ∈ T × {− δ, − δ } , we have ϕ Ax ( t ) > t + τ . Remark . We want to point out that X r and A × id are commutable, thus f r = X r ◦ ( A × id) =( A × id) ◦ X r .Now f r is a chain transitive but nontransitive partially hyperbolic diffeomorphism on T .However, f r is not accessible, since the union of stable and unstable bundles of f r is integrable. Wewill make some more perturbations to achieve the accessibility, and preserving other dynamicalproperties.Let p ∈ T be the fixed point of the linear Anosov automorphism A . Take a small local chart( U ( p ); ( x s , x u )) centered at p in T , such that A ( x s , x u ) = ( λ · x s , λ − · x u ) , x s , x u ) ∈ [ − , s × [ − , u ⊂ U ( p ). Here λ is the eigenvalue of A with 0 < | λ | < < λ − <
10 for the simplicity of symbols. In the rest of this paper, the localcoordinate of ( U ( p ); ( x s , x u )) is the only coordinate we used in T , and we use it in T withoutambiguity. Remark . We want to point out that here we require the neighborhood U ( p ) to be chosenvery small, such that for any point (0 , x u ) with x u = 0, there exists some n >
0, such that A n (0 , x u ) / ∈ U ( p ). The same holds for ( x s ,
0) with x s = 0, and its negative iterations of A .Now we define a C ∞ -smooth function α : T → [0 , α ( x ) = , x ∈ [ − , s × [ − , u ⊂ U ( p ) , , x ∈ T \ [ − , s × [ − , u , ∈ (0 , , otherwise . The function α will help us to prescribe the perturbation region. And the next function γ isused to show the way of perturbations.Let γ : S = [ − , / {− } → R be a C ∞ -smooth function, such that γ ( t ) : (cid:26) > , t ∈ [ − , − τ ) ∪ ( − τ, τ ) ∪ (1 − τ, , = 0 , t ∈ [ − τ, − τ ] ∪ [ τ, − τ ] . We define a smooth vector field Y on T by Y ( x, t ) = − α ( x ) γ ( t ) · ∂∂t , ∀ ( x, t ) ∈ T = T × S . Recall that τ < δ/
2, and so for ρ > ρ map Y ρ satisfies the followingproperties(see Figure 2): • Y ρ ( x, t ) = ( x, ψ x ( t )), and Y ρ ( x, t ) = ( x, t ) for every ( x, t ) ∈ [ − , s × [ − , u × S . • For i = 0 , ψ x ( i ) ≤ i for every x ∈ T . More precisely, for i = 0 , – ψ x ( i ) = i , for every x ∈ [ − , s × [ − , u ; – ψ x ( i ) < i , for every x ∈ T \ [ − , s × [ − , u . • For every ( x, t ) ∈ T × ([ − τ, − τ ] ∪ [ τ, − τ ]), we have Y ρ ( x, t ) = ( x, t ). In particularly,for every x ∈ T and t ∈ {− δ, − δ } , ψ x ( t ) = t .Now we can considering the perturbation of f r made by Y ρ , and it is the diffeomorphism wepromised in our main theorem. Proposition 2.4.
The diffeomorphism f = Y ρ ◦ f r : T → T satisfies the following properties:1. f is a partially hyperbolic skew-product diffeomorphism: f ( x, t ) = ( Ax, ψ Ax ◦ ϕ Ax ( t )) , ∀ ( x, t ) ∈ T .
2. When restricted in the fixed fiber S p , f | S p has two fixed points P , P , and is chain transitive.3. For i = 0 , , ψ Ax ◦ ϕ Ax ( i ) ≤ i for every x ∈ T . More precisely, for i = 0 , , • ψ Ax ◦ ϕ Ax ( i ) = i , for every x ∈ [ − λ − , λ − ] s × [ − λ, λ ] u . P P S p Y ( x, t ) = − α ( x ) γ ( t ) · ∂∂t [ − , s × [ − , u × S U ( p ) × S Figure 2: The perturbation made by Y ρ . • ψ Ax ◦ ϕ Ax ( i ) < i , for every x ∈ T \ [ − λ − , λ − ] s × [ − λ, λ ] u .4. For t ∈ {− δ, − δ } , ψ Ax ◦ ϕ Ax ( t ) > t + τ for every x ∈ T .Proof. the first item comes from the skew-product structure of Y ρ and f r . The second item fromthe vector field Y vanishes in a neighborhood of S p . The third item comes from the fact that f r preserves two tori T and T invariant, and the second property of Y ρ . The last item holdsbecause ψ Ax ( t ) = t for every x ∈ T and t ∈ {− δ, − δ } . f Now we can proof the main theorem from the following three lemmas.
Lemma 3.1.
The diffeomorphism f : T → T is chain transitive.Proof. From the first and second properties of f in Proposition 2.4, we know that f is a partiallyhyperbolic skew-product diffeomorphism on T , thus the stable and unstable manifolds of thefixed fiber S p are dense on T . Since f | S p is chain transitive, this implies f is chain transitive on T . Lemma 3.2.
The diffeomorphism f : T → T is accessible.Proof. Since f is a partially hyperbolic skew-product diffeomorphism on T , if f is not accessible,then from theorem 1.6 of [9], f has a compact us -leaf. Here us -leaf is a compact complete 2-dimensional submanifold which is tangent to E ss ⊕ E uu of f . It is a torus transverse to the5 -fiber of T . Since the compact us -leaf is saturated by W ss and W uu , it intersects every S -fiber of T . Moreover, this us -leaf must intersect every S -fiber with only finitely many points,which comes from it is a compact and complete submanifold.If this compact us -leaf is not periodic by f , then theorem 1.9 of [9] shows that f is semi-conjugated to A times an irrational rotation on S , this implies f has no periodic points. Thiscontradicts to P and P are two fixed points of f , thus f must have a periodic compact us -leaf T us .From the periodicity of T us , we know that T us ∩ S p only contains P or P , and f ( T us ) = T us .Assuming P ∈ T us , then from Theorem 1.7 of [9], we have T us = W ss ( P , f ) = W uu ( P , f ) . In particularly, W ss ( P , f ) and W uu ( P , f ) has strong homoclinic intersections.Recall that from the construction of f , W uu ( P , f ) ∩ { s } × [ − , u × S = { s } × [ − , u × { } . Since W uu ( P , f ) = ∪ n> f n ( W uuloc ( P , f )) and U ( p ) is very small(remark 2.3), property 3 ofProposition 2.4 implies for every ( x, t ) ∈ W uu ( P , f ) \ { s } × [ − , u × S , we have t <
0. Onthe other hand, from property 4 of Proposition 2.4, for every ( x, t ) ∈ W uu ( P , f ), we know that t > − δ + τ and hence − δ + τ < t ≤ W ss ( P , f ) ∩ [ − , s × { u } × S = [ − , s × { u } × { } , and W ss ( P , f ) = ∪ n> f − n ( W ssloc ( P , f )). From the construction of f , for every ( x, t ) ∈ W ss ( P , f ),we have 0 ≤ t < − δ . This implies W ss ( P , f ) ∩ W uu ( P , f ) = { P } , which is a contradiction. The same argument works for P ∈ T us , thus f must be accessible. Lemma 3.3.
The diffeomorphism f : T → T is not transitive.Proof. From the proof of last lemma, we know that W uu ( P , f ) ⊂ T × [ − δ,
0] and W uu ( P , f ) ⊂ T × [1 − δ, f -invariant u -saturated setsΛ i = W uu ( P i , f ) ⊂ T × [ i − δ, i ] , i = 0 , . Notice that Λ i intersects every center leaf.Now we choose two open sets U ⊂ T × ( − , − δ ) and V ⊂ T × (0 , − δ ). Then wemust have f n ( U ) ∩ V = ∅ for every n >
0. Otherwise, there exists Q = ( q, t ) ∈ U and f k ( Q ) = ( A k q, t ′ ) ∈ V for some k >
0. Moreover, there exists some point R = ( A k q, t ′ + s ) ∈ Λ ,such that 0 < s < − δ , and the center interval [ f k ( Q ) , R ) does not intersect Λ .However, there exists some 0 < s < − t , such that the point Q ′ = ( q, t + s ) ∈ Λ , and { q } × [ t, s ) ∩ Λ = ∅ . From the invariance of Λ and Λ , and f preserves the orientation of S -fiber, for every n >
0, the center curve started from f n ( Q ) will meet f n ( Q ′ ) ∈ Λ , and thecenter interval [ f n ( Q ) , f n ( Q ′ )) does not intersect Λ . This is a contradiction for n = k . Thisproves f is not transitive. 6 emark . Actually, the same idea we can prove the following generalized statement. Let f : T → T be a partially hyperbolic skew-product diffeomorphism. If f preserves the orientation ofcenter foliation, and has two disjoint invariant compact u -saturated sets, then f is not transitive.In particularly, if f is transitive, then it has only one minimal u -saturated. In a similar spirit,[10] shows that every partially hyperbolic diffeomorphism on the nonabelian 3-nilmanifolds hasonly one minimal u -saturated set. References [1] F. Abdenur, C. Bonatti, L.J. D´ıaz, Non-wandering sets with non-empty interiors,
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