Measure-Expansive Homoclinic Classes for C 1 Generic Flows

: In this paper, we prove that for a generically C 1 vector ﬁeld X of a compact smooth manifold M , if a homoclinic class H ( γ , X ) which contains a hyperbolic closed orbit γ is measure expansive for X then H ( γ , X ) is hyperbolic.


Introduction and Statements
A homoclinic class H(p, f ) of a diffeomorphism f is the closure of the transverse of homoclinic points associated with a hyperbolic periodic point p. It is an invariant, closed and transitive set. It has a dense orbit and contains a dense set of periodic points which is related to a basic set (see [1]). Many people are paying attention to the study of the homoclinic class for dynamical systems. In fact, the relation between homoclinic classes and expansiveness has been studied by many people. They used various types of C 1 perturbations (C 1 robustly [2,3], C 1 persistently [4], C 1 stably [5], R-robustly [6][7][8] and C 1 generic [9][10][11][12][13][14], etc.). In this paper, we focus on the C 1 generic property. Yang and Gan [14] proved that every expansive homoclinic class of a C 1 generic diffeomorphism of a compact smooth manifold M is hyperbolic. Morales [15] introduced a general notion of expansiveness which is called measure-expansive. Koo, Lee and Lee [16] proved that if every locally maximal homoclinic class H(p, f ) is measure expansive then it is hyperbolic. Lee proved in [6] that if a homoclinic class H(p, f ) is R-robustly measure-expansive, then it is hyperbolic. Later, Lee [13] proved that C 1 generically, a homoclinic class H(p, f ) is expansive if and only if a homoclinic class H(p, f ) is measure-expansive.
About the results of diffeomorphisms, we consider the vector fields which is an extended version of diffeomorphisms.
For vector fields, Bautista [17] showed that the geometric Lorenz attractor is a homoclinic class which contains a singular point. Komuro showed in [18] that the geometric Lorenz attractor is K * -expansive. However, the geometric Lorenz attractor is not expansive (see [19]). Nevertheless, in vector fields, the relation betwwen homoclinic classes and expansiveness are still interesting research subjects. Lee and Park [20] proved that every expansive locally maximal homoclinic class H(γ, X) of C 1 generic vector fields is hyperbolic. Lee and Oh [21] proved that every measure-expansive locally maximal homoclinic class H(γ, X) of C 1 generic vector fields is hyperbolic. The results used a dynamical condition which is locally maximal. In [10], Lee proved that if a homoclinic class H(γ, X) is R-robustly measure-expansive then it is hyperbolic.
We study the hyperbolicity of a measure-expansive homoclinic class for C 1 generic vector fields without the locally maximal condition and R-robust property, which is a generalization of previous results.

Basic Definitions and Main Theorem
Assume that M is a compact smooth Riemannian manifold. Denote by X(M) the set of a C 1 vector fields on M, endowed with then C 1 topology. Every vector field X generates a C 1 flow X t : M → M, t ∈ R. Denote by Sing(X) = {x ∈ M : X t (X) = x for all t ∈ R} the set of singularities of X, and by Per(X) = {x ∈ M : there is π(x) > 0 such that X π(x) (x) = x} the set of periodic poins of X. A point x ∈ M is said to be critical point if x ∈ Sing(X) ∪ Per(X), and denote by Crit(X) = Sing(X) ∪ Per(X). Let H be the set of all continuous maps h : R → R with h(0) = 0. A vector field X ∈ X(M) is said to be expansive if for every > 0 there are δ > 0 and h ∈ H such that if for any x, y ∈ M and d(X t (x), X h(t) (y)) ≤ δ ∀t ∈ R, then y = X t 0 (x) for some |t 0 | < . Bowen and Walters [22] proved that if a vector field X ∈ X(M) is expansive then every singular points is isolated. Oka [23] proved that if a vector field X ∈ X(M) is expansive, then Sing(X) = ∅.
We define the following It is said to be the δ-dynamic ball of X.
Then we can rewrite the δ-dynamic ball such as The following concept which is a general notion of expansiveness for flows was defined by Carrasco-Olivera and Morales [24].
Definition 1. Let X ∈ X(M) and let Λ ⊂ M be a closed X t -invariant set. We say that Λ is measure-expansive if Λ is µ-expansive of X, for any µ ∈ M * (M). If Λ = M, then X is called measure-expansive.
is one dimensional and generated by the vector field X, there are constants C > 0 and 0 < λ < 1 so that for every x ∈ Λ and t ≥ 0, Let γ ∈ Per(X) be hyperbolic. We say that η ∈ Per(X) is homoclinically related to γ ∈ Per(X), Then we define it as H(γ, X) = {η ∈ Per(X) : η ∼ γ}.
In the paper, we consider the homoclinic class H(γ, X) which contains a hyperbolic saddle type of the periodic orbit γ. The following is the main theorem of this paper.

Theorem 1.
There is a residual set G δ ⊂ X(M) such that for any x ∈ G δ , if a homoclinic class H(γ, X) is measure-expansive of X, then it is hyperbolic.

Proof of Theorem 1
Let M be as before and let X ∈ X(M).
The following was proved by [25], where it is called a vector field version of Franks lemma.

Lemma 2.
Let p belongs to a periodic orbit for X with period π(p) > 0, and let f : N p,r 1 → N p is the Poincaré map of X(for some r 1 > 0). Let 0 < r ≤ r 1 be given. Then there are a positive δ > 0 and 0 < 0 < r/2 such that for a linear isomorphism L : N p → N p with L − D p f < δ, there exists Y C 1 closed to X with the properties: p belongs to a periodic orbit for Y, Here, g : N p,r → N p is the Poincaré map of Y.
A closed orbit γ is weakly hyperbolic if for any p ∈ γ and any > 0, D p f has an eigenvalue µ such that (1 − ) < |µ| < (1 + ), where f is the Poincaré map of X. Denote by W H(X) the set of all weak hyperbolic periodic orbits of X.

Lemma 3.
Let η ∈ H(γ, X) ∩ Per(X) with η ∼ γ. If η ∈ W H(X) then there is Y C 1 close to X such that g has a small arc J with the endpoints are homoclinically related to p ∈ γ Y , where g is the Poincaré map of Y and γ Y is continuation of γ.
Proof. Let η ∈ H(γ, X) ∩ Per(X) with η ∼ γ. Suppose that η ∈ W H(X). Take q ∈ η and let f : N q,r → N q (for some r > 0) be the Poincaré map of X. Since η is a weak hyperbolic periodic orbit of X, according to Lemma 2 there exist δ(= ) > 0 and 0 < 0 < r/2 such that for a map L : where, g : N q,r → N q is the Poincaré map of Y. Then we have that λ is an eigenvalue of D q g with |λ| = 1. Take a vector u( = 0) which is associated to the eigenvalue λ and u = 0 /4. Then we obtain that g(exp q (u)) = exp q • L • exp −1 q (exp q (u)) = exp q (u).
Let J = {tu : − 0 /4 ≤ t ≤ 0 /4} and let J = exp q (J). Then g k | J is the identity, for some k > 0, and g i (x) goes to W s loc (q) for x ∈ J and i ≥ 0. Let q 1 and q 2 be the endpoints of J . Then as in the proof of ( [4], Proposition 3), there is h C 1 close to g (also C 1 close to f ) such that q 1 , q 2 and q are the only periodic points of h, h k | J 1 is the identity and q 1 ∼ p h ∈ γ Z and q 2 ∼ p h ∈ γ Z , where γ Z is the continuation of γ and the Poincaré map h is associated with a vector field Z which is C 1 close to Y. Thus if η ∈ W H(X) then there is Z C 1 close to X such that h has a small arc J 1 with the endpoints q 1 and q 2 are homoclinically related to p h ∈ γ Z , where h is the Poincaré map of Z. This proves Lemma 3.
Let f : N p,r → N p be the Poincaré map of X ∈ X(M). The following is a vector field version of what introduced by Yanf and Gan in [14]. For any > 0, a closed small arc I is -periodic curve if  There is a residual set G 1 ⊂ X(M) such that for any X ∈ G 1 , any hyperbolic periodic orbit γ of X, and given > 0, if any C 1 neighborhood U (X) of X, there is Y ∈ U (X) ⊂ X(M) which has an -periodic curve J for which the two endpoints of J are homoclinically related to γ Y then X has a 2 -periodic curve L for which the two endpoints of L are homoclinically related to γ (see [14]).
For any x, y ∈ M, and δ > 0, a sequence {x 0 (= x), x 1 , x 2 , . . . , We say that y is chain-attainable from x if for any δ > 0, there is a δ-chain from x and y. A point x ∈ M is a chain recurrence point if x is chain-attainable from itself. Denote by CR(X) the set of all chain recurrence points of X. Note that chain bi-attainability is closed equivalence relation in CR(X). For any x ∈ CR(X), the equivalent class which has x is said to be the chain recurrence class of x, and we denote by C(x, X) or C(Orb(x), X).

Lemma 5.
There is a residual set G 2 ⊂ X(M) such that for any X ∈ G 2 , (a) X is Kupka-Smale, that is, any critical orbit is hyperbolic and W s (σ) is transverse to W u (η), where σ and η are critical orbits of X (see [26]).

Lemma 6.
There is a residual set G 3 ⊂ X(M) such that for any X ∈ G 3 , if H(γ, X) is measure-expansive then for any η ∈ H(γ, X) ∩ Per(X) with η ∼ γ, Proof. Let X ∈ G 3 = G 1 ∩ G 2 , and let H(γ, X) be measure-expansive. Suppose that there is η ∈ H(γ, X) ∩ Per(X) with η ∼ γ such that η ∈ H(γ, X) ∩ W H(X). Take p ∈ η such that p is a weak hyperbolic point of X. According to Lemma 3, for any δ > 0 there is Y C 1 close to X such that g has a small arc J with the endpoints are homoclinically related to p Y ∈ γ Y which is a δ-periodic curve, and g k | J : J → J is the identity map for some k > 0, where g is the Poincaré map of Y and γ Y is continuation of γ. For simplicity, we assume that g k | J = g| J . It is known that J ⊂ C(p Y , g) ⊂ C(γ Y , Y). By Lemma 4, one can see that X has a 2 -periodic curve L for which the two endpoints of L are homoclinically realte to γ. By Lemma 5 (b), f has a 2 -periodic curve L ⊂ H(γ, X) = C(γ, X). Let ν be the normalized Lebesgue measure on L. Define a measure χ on M by for some Borel set B of M. It is clear that χ(B) = 0. Take = δ, and let x ∈ L. Then we define Γ(x, ) = {y ∈ M : d( f i (x), f i (y)) ≤ for all i ∈ Z}. Let Φ(x, ) = {z ∈ L : d( f i (x), f i (y)) ≤ for all i ∈ Z}. It is clear that Φ(x, ) ⊂ Γ(x, ). Since H(γ, X) is measure-expansive, we know that χ(Γ(x, )) = 0. Since Φ(x, ) ⊂ Γ(x, ), χ(Φ(x, )) should be 0. This is a contradiction since χ(Φ(x, )) = 0.
Let p ∈ γ be a hyperbolic periodic point of X with the period π(p), and let f : N p,r → N p be the Poincaré map with respect to X. Then if µ 1 , µ 2 , . . . , µ n−1 are the eigenvalues of D p f , then for i = 1, 2, . . . , n − 1 are called the Lyapunov exponents of p.
Denote by H * (γ, X) = {x ∈ M : x ∈ H(γ, X) and x ∈ Sing(X)}. For vector fields, we assume that H * (γ, X). Wang proved in [28] that a vector field X in a dense G δ subset of X(M), if a homoclinic class H * (γ, X) is not hyperbolic, then one can find a periodic orbit ζ of X that is homoclinically related to γ and has a Lyapunov exponent arbitrarily close to 0.
Hereafter, we say that a property holds for C 1 generic vector fields if it is satisfied on a dense G δ subset of X(M). Note that for a C 1 generic vector field X, if a periodic orbit γ has a Lyapunov exponent arbitrarily close to 0, then one can take a periodic orbit ζ such that ζ ∈ W H(X).
Note that if H(γ, X) ∩ Sing(X) = ∅ then the result of Wang [28] is not true. Indeed, the geometric Lorenz attractor is an example of that. However, we consider H * (γ, X), then, we can rewrite the result of Wang [28] as follows.

Lemma 7.
There is a residual set G 4 ⊂ X(M) such that for any X ∈ G 4 , if a homoclinic class H * (γ, X) is not hyperbolic, then there is a periodic orbit η ∈ H * (γ, X) such that η ∼ γ and η ∈ W H(X).
Since an expansive flow is a measure-expansive flow, according to Theorem 1, C 1 generically, if a homoclinic class H(γ, X) which contains a hyperbolic periodic orbit γ is expansive, then H(γ, X) is hyperbolic. Thus we have the following, which is a generalization of the result of [20]. Corollary 1. For C 1 generic X ∈ X(M), a homoclinic class H(γ, X) which contains a hyperbolic periodic orbit γ is expansive if and only if H(γ, X) is measure-expansive.

Conclusions
The paper considers the relationship between measure-expansiveness and homoclinic classes for flows. More in detail, we proved that there is an open and dense G δ in X(M) for any X ∈ G δ , if a homoclinic class H(γ, X) is measure-expansive then H(γ, X) is hyperbolic for X. The results are an extension and generalization of the previious results (see [6,8,13,14,16,20,21]).