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Continuum-wise expansive diffeomorphisms and conservative systems

Abstract

We prove that C 1 -generically, continuum-wise expansive diffeomorphisms satisfy both Axiom A and the no-cycle condition. Moreover, (i) if a volume-preserving diffeomorphism belongs to the C 1 -interior of the set of all continuum-wise expansive volume-preserving diffeomorphisms then it is Anosov, and (ii) C 1 -generically, every continuum-wise expansive volume-preserving diffeomorphism is transitive Anosov.

MSC:37C20, 37D20.

1 Introduction

Let Diff(M) be the space of diffeomorphisms of closed C -manifolds M endowed with the C 1 -topology, and let d denote the distance on M induced from a Riemannian metric on the tangent bundle TM. In dynamical systems, expansivity is a useful notion to study of the stability. Roughly speaking, if two points stay near for future and past iterates, then they must be equal. We say that f is expansive if there is e>0 such that for any pair of distinct points x,yM, d( f n (x), f n (y))>e for some nZ. The number e>0 is called an expansive constant for f.

For a point xM, we say that x is a non-wandering point if for any neighborhood U of x, there is nZ such that f n (U)U. Denote by Ω(f) the set of all non-wandering points of f. It is clear P ( f ) ¯ Ω(f), where P(f) is the set of the periodic points of f, and P ( f ) ¯ is the closure of P(f). We say that f satisfies Axiom A if Ω(f)= P ( f ) ¯ is hyperbolic. We say that f is quasi-Anosov if for any vTM (v0) the set {D f n (v):nZ} is unbounded. It follows that f satisfies Axiom A.

For expansivity, in [1], Mañé showed that a diffeomorphism belongs to the C 1 -interior of the set of all expansive diffeomorphisms if and only if f is quasi-Anosov.

In this paper, we study the notion of continuum-wise expansivity which was introduced by Kato in [2]. Let Λ be a closed set of M. A set Λ is nondegenerate if the set Λ is not reduced to one point. We say that ΛM is a subcontinuum if it is a compact connected nondegenerate subset Λ of M. A diffeomorphism f on M is said to be continuum-wise expansive if there is a constant e>0 such that for any nondegenerate subcontinuum A there is an integer n=n(A) such that diam f n (A)e, where diamS=sup{d(x,y):x,yS} for any subset S of M. Such a constant α is called a continuum-wise expansive constant for f. Note that every expansive homeomorphism is continuum-wise expansive diffeomorphism, but its converse is not true (see [[3], Example 3.5]). For diffeomorphisms, we introduce an example. It is well known that S 2 does not admit an expansive diffeomorphism, but it admits a continuum-wise expansive diffeomorphisms (see [4]).

2 Continuum-wise diffeomorphisms

Let M be as before, and let fDiff(M). Denote by E(M) and CWE(M) the set of all expansive diffeomorphisms and the set of all continuum-wise expansive diffeomorphisms, respectively. Sakai [5] proved that fCWE(M) if and only if the diffeomorphism is quasi-Anosov. By Mañé’s result [1], we know the following.

Theorem 2.1 The C 1 -interior of CWE(M) coincides with the C 1 -interior of E(M).

We say that Λ is transitive set if there is a point xΛ such that ω f (x)=Λ, where ω f (x) is the ω-limit set of x. Let ΛM be an f-invariant closed set. We say that Λ admits a dominated splitting if the tangent bundle T Λ M has a continuous Df-invariant splitting EF and there exist constants C>0 and 0<λ<1 such that

D x f n | E ( x ) D x f n | F ( f n ( x ) ) C λ n

for all xΛ and n0. Recently, Lee [6] showed that if a transitive set Λ is C 1 -stably continuum-wise expansive then it admits a dominated splitting.

A subset RDiff(M) is called residual if it contains a countable intersection of open and dense subsets of Diff(M). A dynamic property is called C 1 -generic if it holds in a residual subset of Diff(M). We use the terminology for C 1 -generic f to express there is a residual subset RDiff(M), and fR.

Recently, in [7], Arbieto proved that for C 1 -generic fDiff(M), f is expansive then f is Ω-stable, that is, obeys Axiom A and the no-cycle condition. We stated the above fact.

Theorem 2.2 For C 1 -generic f, if f is expansive then f satisfies both Axiom A and the no-cycle condition.

In this spirit, we show that C 1 -generically, every continuum-wise expansive diffeomorphism satisfies both Axiom A and the no-cycle condition. This is a generalization of the remarkable result in [7].

Theorem A For C 1 -generic f, if f is continuum-wise expansive then f satisfies both Axiom  A and the no-cycle condition.

3 Continuum-wise volume-preserving diffeomorphisms

Let M be a closed C Riemannian manifold endowed with a volume form ω. Let μ denote the Lebesgue measure associated to ω, and let d denote the metric induced on M by the Riemannian structure. Denote by Diff μ (M) the set of diffeomorphisms which preserves the Lebesgue measure μ endowed with the Whitney C 1 -topology. Note that in volume-preserving diffeomorphisms, the non-wandering set Ω(f)=M by recurrent theorem. We say that Λ is hyperbolic if the tangent bundle T Λ M has a Df-invariant splitting E s E u and there exist constants C>0 and 0<λ<1 such that

D x f n | E x s C λ n and D x f n | E x u C λ n

for all xΛ and n0. Moreover, if Λ=M then f is Anosov. Note that f is Anosov then f is expansive, and so, f is continuum-wise expansive. In [8], Bessa et al. proved that a volume-preserving diffeomorphism belongs to the C 1 -interior of the set of all expansive volume-preserving diffeomorphisms if and only if it is Anosov. For the another conservative cases, that is, geodesic flow and a Hamiltonian system, Bessa et al. have shown in [9] that if a Hamiltonian system belongs to the C 2 -interior of the set of all expansive Hamiltonian systems then it is Anosov. And Ruggiero [10] showed that if a geodesic flow belongs to the C 1 -interior of the set of all expansive geodesic vector fields then it is Anosov.

Let CWE μ (M) be the set of all continuum-wise expansive volume-preserving diffeomorphisms. In this paper, we study the continuum-wise expansive case, and if f belongs to the C 1 -interior of CWE μ (M), then f is Anosov. Let int CWE μ (M) denote the C 1 -interior of the set of all continuum-wise expansive volume preserving diffeomorphisms. In this paper, we prove the following theorem.

Theorem B The set AN μ (M) of Anosov diffeomorphisms in Diff μ (M) coincides with the C 1 -interior of the set of continuum-wise expansive diffeomorphisms in Diff μ (M); that is, AN μ (M)=int CWE μ (M).

In diffeomorphisms, Arbieto [7] proved that C 1 -generically, if f is expansive then f is Ω-stable. It is well known that for a Ω-stable diffeomorphism, there is a diffeomorphism such that the diffeomorphism is not expansive. However, for volume-preserving diffeomorphisms, the phenomenon cannot happen since Ω(f)=M. In dimM=2, for C 1 -generic f, if a C 1 -neighborhood U(f) of f, there is gU(f) such that g has a periodic point p g with homoclinic tangency q g then f has a periodic point p with homoclinic tangency q. In fact, it is closely related to the conjecture of Smale (see [11]). Note that if dimM=2 then it does not exist normally hyperbolic. In this paper, we consider dimM3. Recently, Bessa et al. [8] proved that dimM3, for C 1 -generic f, if f Diff μ (M) is expansive then f is Anosov. For a Hamiltonian system, Lee [12] showed that C 2 -generically, an expansive Hamiltonian system is Anosov. In this spirit, we study the continuum-wise expansiveness for generic view point. Then we have the following.

Theorem C For C 1 -generic f, if f is continuum-wise expansive then it is transitive Anosov.

4 Proof of Theorem A

Let dimM3 and let fDiff(M). We prepare several lemmas to arrive at Theorem A. The Franks lemma [13] will play an essential role in our proofs.

Lemma 4.1 Let U(f) be any given C 1 -neighborhood of f. Then there exist ε>0 and a C 1 -neighborhood U 0 (f)U(f) of f such that for given g U 0 (f), a finite set { x 1 , x 2 ,, x N }, a neighborhood U of { x 1 , x 2 ,, x N } and linear maps L i : T x i M T g ( x i ) M satisfying L i D x i gε for all 1iN, there exists g ˆ U(f) such that g ˆ (x)=g(x) if x{ x 1 , x 2 ,, x N }(MU) and D x i g ˆ = L i for all 1iN.

Let p be a periodic point of f, and let 0<δ<1. We say p has a δ-weak eigenvalue if D p f π ( p ) has an eigenvalue λ such that ( 1 δ ) π ( p ) <|λ|< ( 1 + δ ) π ( p ) . The following lemma will also play a crucial role in our proof.

Lemma 4.2 [[7], Lemma 5.1]

There exists a residual set R 1 Diff(M) such that for any f R 1 ,

  1. (1)

    for any δ>0, if for any C 1 -neighborhood U(f), there is gU(f) which has a hyperbolic p g P(g) with a δ-weak eigenvalue, then f has a hyperbolic point pP(f) with a 2δ-weak eigenvalue;

  2. (2)

    for any δ>0, if f has a hyperbolic point qP(f) with a δ-weak eigenvalue, then f has a hyperbolic point pP(f) with a δ-weak eigenvalue, whose eigenvalues are all real.

Remark 4.3 If f has a normally hyperbolic, then by Hirsh et al. [14] and Mañé [15], it is C 1 -robust, that is, for any g C 1 -close to f, g has a normally hyperbolic then f also has a normally hyperbolic (see also [16]).

Lemma 4.4 There exists a residual set R 2 Diff(M) such that for f R 2 if f is continuum-wise expansive, then there exists δ>0 such that f has no δ-weak eigenvalue.

Proof Let R 2 = R 1 , and let f R 2 be continuum-wise expansive for f. Suppose, by contradiction, that for any δ>0 there is a periodic point p of f such that p has a δ-weak eigenvalue. Let ε>0, and let V(f) U 0 (f) be a C 1 -neighborhood of f which is given by Lemma 4.1 with respect to U 0 (f). Then there exist gU(f) and a non-hyperbolic periodic point q of g such that an eigenvalue λ of D q g π ( q ) with |λ|=1, and T q M= E c (q) E s (q) E u (q), where E σ (q), σ=c,s,u, are D q g π ( q ) -invariant subspaces corresponding to eigenvalues λ of D q g π ( q ) for |λ|=1, |λ|<1, and |λ|>1, respectively. Let W(f)V(f) be the C 1 ε 0 -ball of f. Set C= sup x M { D x g}. For 0< ε 1 < ε 0 , we can obtain a linear automorphism O: T q M T q M such that

  1. (i)

    Oid< ε 1 C ,

  2. (ii)

    keeps E σ invariant, where σ=c,s,u,

  3. (iii)

    all eigenvalues of O D q g π ( q ) , say μ j , j=1,2,,c, are roots of unity.

Let F be the finite set {q,g(q),, g π ( q ) 1 (q)}. Define

L j = { D g j ( q ) g , j = 0 , 1 , , π ( q ) 2 , O D g π ( q ) 1 ( q ) g , j = π ( q ) 1 .

Observe that L π ( q ) 1 D g π ( q ) 1 ( q ) gOid D g k 1 ( q ) g< ε 0 . Thus L j D g j ( q ) g< ε 0 for all j=0,1,,π(p)1. By Lemma 4.1, we can find a diffeomorphism g 1 W(f) and δ 0 >0 such that

  1. (a)

    B 4 δ 0 ( g i (q)) B 4 δ 0 (q)=, 0ijπ(q)1,

  2. (b)

    g 1 =g on F(M j = 0 π ( q ) 1 B 4 δ 0 ( g j (q))),

  3. (c)

    g 1 = exp g j + 1 ( q ) L j exp g j ( q ) 1 on B δ 0 ( g j (q)), 0jπ(q)1.

Define

L=O D q g π ( q ) = j = 0 π ( q ) 1 L j ,

where B δ (p) denotes the δ-neighborhood of p.

Then by (iii) we can find m>0 such that L m | E c ( q ) =id | E c ( q ) . Choose a small δ 1 satisfying 0<4 δ 1 < δ 0 such that

L m k ( T q M ( 4 δ 1 ) ) T q M( δ 0 ),

where T q M( δ 1 )={v T q M|v δ 1 }. Then by (c) we have

( g 1 π ( q ) ) m = g 1 m π ( q ) = exp q G m exp q 1

on exp q ( T q M(4 δ 1 )).

We write

T q M( δ 1 )= E c (q, δ 1 ) E s (q, δ 1 ) E u (q, δ 1 ),

where E σ (q, δ 1 )= E σ (q) T q M( δ 1 ), σ=c,s,u. Then exp q ( E c (q,4 δ 1 )) is ( g 1 k ) m -invariant. Since f R 1 , we assume that the eigenvalue λR.

Put exp q ( E c (q,4 δ 1 )) is an arc I q centered at q. Observe that ( g 1 k ) m =id on exp q ( E c (q,4 δ 1 )). By our construction, ( g 1 k ) m is the identity on the arc I q . It is clear that the small arc I q is normally hyperbolic for g 1 . By Remark 4.3, for any g C 1 -close to f, if g has a normally hyperbolic then f has a normally hyperbolic, that is, it is C 1 -robust. Then we know that f has a small arc J q which centered at q with f π ( q ) ( J q )= J q . Note that if f is continuum-wise expansive then f k is continuum-wise expansive for any kZ (see [[2], Proposition 2.6]). Denote by l(A) the length of A. Take e=2l( J q ). Since J q is f π ( q ) -invariant, for all nZ,

diam ( f n ( J q ) ) <e.

This is a contradiction. □

We say that f satisfies star condition if there is a C 1 -neighborhood U(f) such that for any gU(f), every pP(g) is hyperbolic. We denote by F(M) the set of diffeomorphisms satisfying star condition.

Lemma 4.5 There is a residual set R 3 Diff(M) such that for any continuum-wise expansive map f R 3 , fF(M).

Proof Let R 3 = R 2 , and let f R 3 be continuum-wise expansive. Proof by contradiction, we may assume that fF(M). Then by Lemma 5.1, there is g C 1 -close to f and p g P(g) such that for any δ>0, p g has a δ/2-weak eigenvalue. By Lemma 4.2, pP(f) has a δ-weak eigenvalue. This is a contradiction by Lemma 4.4. □

Proof of Theorem A Let f R 3 be continuum-wise expansive. By Lemma 4.5, fF(M). Since fF(M), By Aoki [17] and Hayashi [18], we know that f satisfies both Axiom A and the no-cycle condition. Thus it is Ω-stable. □

5 Proof of Theorem B and Theorem C

Let M and let f Diff μ (M) be as before. To prove our result, we use the Franks lemma, which is proved in [[19], Proposition 7.4].

Lemma 5.1 Let f Diff μ 1 (M), and be a C 1 -neighborhood of f in Diff μ 1 (M). Then there exist a C 1 -neighborhood U 0 U of f and ε>0 such that if g U 0 , any finite f-invariant set E={ x 1 ,, x m }, any neighborhood U of E and any volume-preserving linear maps L j : T x j M T g ( x j ) M with L j D x j gε for all j=1,,m, there is a conservative diffeomorphism g 1 U coinciding with f on E and out of U, and D x j g 1 = L j for all j=1,,m.

We denote by F μ (M) the set of diffeomorphisms f Diff μ (M) which have a C 1 -neighborhood U(f) Diff μ (M) such that for any gU(f), every periodic point of g is hyperbolic.

Very recently, Arbieto and Catalan [20] proved that every volume-preserving diffeomorphism in F μ (M) is Anosov.

Theorem 5.2 [[20], Theorem 1.1]

Every diffeomorphism in F μ (M) is Anosov.

To prove Theorem B, it is enough to show that a continuum-wise expansive volume-preserving diffeomorphism f F μ (M).

Remark 5.3 Let f Diff μ 1 (M). From the Moser theorem (see [21]), we can find a smooth conservative change of coordinates φ x :U(x) T x M such that φ x (x)=0, where U(x) is a small neighborhood of xM.

Lemma 5.4 If fint CWE μ (M), then f F μ (M).

Proof Take fint CWE μ (M), and U(f) a C 1 -neighborhood of f. Let ε>0 and V(f)U(f) be corresponding number and C 1 -neighborhood given by Lemma 5.1. To derive a contradiction, suppose that there is a non-hyperbolic periodic point pP(g) for some gV(f). To simplify the notation in the proof, we may assume that g(p)=p. Then there is at least one eigenvalue λ of D p g such that |λ|=1. By making use of Lemma 5.1, we linearize f at p with respect to Moser’s theorem, that is, by choosing α>0 sufficiently small we construct g 1 C 1 -nearby g such that

g 1 (x)= { φ p 1 D p g φ p ( x ) if  x B α ( p ) , g ( x ) if  x B 4 α ( p ) .

Then g(p)= g 1 (p)=p. Thus T p M= E c E σ , where E p c associated to λ=1 and E p σ associated to eigenvalues less than one and greater than one. Take η=α/4. Then we define E c (η) φ p ( B α (p))= E c (η).

Case 1. dim E p c =1.

Since p is non-hyperbolic for g 1 , by our construction, we may assume that there is l>0 such that D p g 1 l (v)=v for any v E p c (η) φ p ( B α (p)). Take v E p c (η) such that v=η/4. Then we can find a small arc I p = φ p 1 ({tv:1tη/4}) B α (p) such that

  1. (i)

    g 1 i ( I p ) g 1 j ( I p )= if 0ijl1,

  2. (ii)

    g 1 l ( I p )= I p , that is, g 1 l | I p is the identity map,

  3. (iii)

    I p is normally hyperbolic.

For simplicity, we assume that g 1 l = g 1 . Take e=η. Then for all nZ,

diam ( g 1 n ( I p ) ) <e.

This is a contradiction.

Case 2. dim E p c =2.

In the proof of the second case, to avoid notational complexity, we consider the case g(p)=p. By Lemma 5.1, there is α>0 and hU(f) such that h(p)=g(p)=p and h(x)= φ p 1 D p g φ p (x) if x B α (p). With a small modification of D p g, we may assume that there is l>0 such that D p g l (v)=v for any v E p c (α) by Lemma 5.1. We can choose v E p c (α) such that v=α/4 and we set D p = φ p 1 ({tv:1tα/4}) B α (p). Then the disk D p satisfies the following conditions:

  1. (i)

    h i ( D p h ) h j ( D p h )= if 0ijl1,

  2. (ii)

    h l ( D p h )= D p h , that is, h l | D p h is the identity map,

  3. (iii)

    D p is normally hyperbolic.

As in the proof of the dim E p c =1, we can derive a contradiction. □

Proof of Theorem B Suppose that fint CWE μ (M). By Lemma 5.4, f F μ (M). Thus by Theorem 5.2, f is Anosov. □

Proof of Theorem C The proof of Theorem C is parallel the proof of Theorem A. Indeed, to prove Theorem A we use previous results - Lemmas 4.2, 4.4 and 4.5. Then we have a volume-preserving diffeomorphism f F μ (M). Thus f is Anosov. □

In diffeomorphisms, there is an open problem: are Anosov diffeomorphisms transitive? In [13] Franks and [22] Newhouse proved it for codimension one Anosov diffeomorphisms. It was announced in Xia in a talk, Anosov diffeomorphisms are transitive, an invited talk of the Rocky Mountain Conference on Dynamical Systems, May 12-14, 2008, that every Anosov diffeomorphism is transitive. It has not been published yet. Nevertheless, in the volume-preserving diffeomorphism, an Anosov diffeomorphism has the non-wandering set equal to the whole manifold M by the Poincaré theorem. By the shadowing property of the hyperbolic sets the periodic points are dense in M. And by the Smale spectral decomposition theorem, we have a single piece equal to M, and so, we have transitivity. Thus, the Anosov volume-preserving diffeomorphism is transitive, which is a direct consequence of classic hyperbolic dynamics. But in volume-preserving diffeomorphisms Bonatti and Crovisier proved that C 1 -generically, a volume-preserving diffeomorphism is transitive.

Theorem 5.5 [[23], Theorem 1.3]

There is a residual set R 4 Diff μ (M) such that for any f R 4 , f is transitive and M is a unique homoclinic class.

We say that f is transitive if there is a point xM such that ω(x)=M, where ω(x) is the omega limit set.

Remark 5.6 In [[24], Theorem 1.3], Newhouse showed that C 1 -generic volume-preserving diffeomorphisms in surfaces are Anosov or else the elliptical points, nonreal eigenvalues conjugated and of norm one, are dense.

By [8] and Theorem A, we have the following.

Corollary 5.7 There is a residual set G Diff μ (M) such that for any fG, the following are equivalents:

  1. (a)

    f is expansive,

  2. (b)

    f is transitive Anosov.

    Moreover, if dimM3 then

  3. (c)

    f is continuum-wise expansive,

  4. (d)

    f has the shadowing property,

  5. (e)

    f has the weak specification property.

Proof Let fG= R 3 R 4 is continuum-wise expansive. By Theorem A, f is Anosov. Since f R 4 , by Lemma 5.5, f is transitive. Thus if f is continuum-wise expansive, then f is transitive Anosov. By Bessa et al. [8], f is expansive, then f is Anosov, and so, f is transitive Anosov. If dimM3, then by Bessa et al. [8] if f has the shadowing property and f has the weak specification property, then f is Anosov and since f R 4 , also f is transitive. Thus f is transitive Anosov. □

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Acknowledgements

The author would like to thank the referee for his/her useful comments, detailed corrections and suggestions. This work is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (No. 2014R1A1A1A05002124).

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Lee, M. Continuum-wise expansive diffeomorphisms and conservative systems. J Inequal Appl 2014, 379 (2014). https://doi.org/10.1186/1029-242X-2014-379

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Keywords

  • Axiom A
  • expansive
  • continuum-wise expansive
  • Anosov
  • transitive
  • generic property