CONTINUUM-WISE EXPANSIVE DIFFEOMORPHISMS

In this paper, we show that the C1 interior of the set of all continuum-wise expansive diffeomorphisms of a closed manifold coincides with the C1 interior of the set of all expansive diffeomorphisms. And the C1 interior of the set of all continuum-wise fully expansive diffeomorphisms on a surface is investigated. Furthermore, we have necessary and sufficient conditions for a diffeomorphism belonging to these open sets to be Anosov.

investigate the dynamics of continuum-wise expansive diffeomorphisms from the differentiable viewpoint.
Let M be a C ∞ closed manifold and let Diff(M ) be the space of C 1 diffeomorphisms of M endowed with C 1 topology.Let E(M ) and CE(M ) be the set of all expansive diffeomorphisms and the set of all continuumwise expansive diffeomorphisms in Diff(M ) respectively.Theorem 1.The C 1 interior of CE(M ), int CE(M ), coincides with the C 1 interior of E(M ), int E(M ).
The notion of quasi-Anosov diffeomorphisms on M was introduced by Mañé [9] and it was proved in [8] that f ∈ int E(M ) if and only if f is a quasi-Anosov diffeomorphism.Clearly if f is quasi-Anosov, then f ∈ int CE(M ).Our theorem will be obtained by showing that if f ∈ int CE(M ), then f is quasi-Anosov.
Let (X, d) be as before and assume that X is a continuum.A homeomorphism f : X → X is called continuum-wise fully expansive ( [6]) provided that for every ε > 0 and δ > 0, there is a natural number Here d H denotes the Hausdorff metric.It is clear that every continuume-wise fully expansive homeomorphism is continuume-wise expansive but its converse is not true (see [6,Example 4.9]).Remark that there is an example [6, Example 2.2] of continuum-wise fully expansive homeomorphisms that is not expansive.The notion of continuum-wise fully expansive is closely related to that of topologically mixing.Indeed, it is proved in [6] that if f : X → X is continuum-wise fully expansive, then f is topologically mixing.
Let we denote the set of all continuum-wise fully expansive diffeomorphisms of M by CFE(M ) and denote by int CFE(M ) its C 1 interior.Then we have the following This corollary is an easy consequence of Theorem 1. Indeed, clearly int CFE(M ) ⊂ int CE(M ).By Theorem 1 int CE(M ) is equal to the set of all quasi-Anosov diffeomorphisms.Since every quasi-Anosov diffeomorphism f is Anosov when dim M = 2 (see [9]) and since every Anosov diffeomorphism is topologically conjugate to a hyperbolic toral automorphism when dim M = 2, f is continuum-wise fully expansive (see [6,Proposition 2.4]).
Let M be as before and let d be a metric on M induced from a Riemannian metric We say that f has the shadowing property if for ε > 0 there is δ > 0 such that every δ-pseudo-orbit of f can be ε-shadowed by some point.
Let us denote by H(M ) the set of all homeomorphisms of M endowed with C 0 topology.We say that f is structurally stable if there is a C 1 neighborhood U(f ) ⊂ Diff(M ) such that for every g ∈ U(f ), there is for a sufficently small ε.It is well known that every structurally stable diffeomorphism is topologically stable ( [12]) but its converse is not true, and that every topologically stable diffeomorphism has the shadowing property ( [11]).The author do not know that whether every diffeomorphism having the shadowing property is topologically stable (cf.[14]). We It is easy to see that every topologically stable diffeomorphism f on M is persistent.Notice that every pseudo-Anosov diffeomorphism f on a surface is persistent (see [7, Corollary 3.1]) but f does not have the shadowing property so that f is not topologically stable.These notions are independent of metrics for M and are conjugacy invariant.
Clearly every Anosov diffeomorphism is contained in int CE(M ), however, there exists f ∈ int CE(M ) that is not Anosov when dim M = 3 (see [3]).In this paper, under the above notations, we have the following Theorem 2. Let f ∈ int CE(M ).Then the following conditions are mutually equivalent.

Proofs of Theorems
It was also proved in [9, Theorem A] that f ∈ Diff(M ) is quasi-Anosov if and only if f satisfies Axiom A and for every x ∈ M we have where W s (x) and W u (x) are the stable manifold and the unstable manifold of x ∈ M respectively.Hereafter, let QA(M ) ⊂ Diff(M ) be the set of all quasi-Anosov diffeomorphisms.As we stated before Mañé proved that QA(M ) = int E(M ), and thus, to prove Theorem 1 it is enough to show that int CE(M ) ⊂ QA(M ).
Our proof of this theorem rely largely on the result which was proved by Aoki [1] and Hayashi [4] independently.Let P (f ) denote the set of all periodic points of f ∈ Diff(M ), and let F(M ) be the set of all f ∈ Diff(M ) having a C 1 -neighborhood U(f ) ⊂ Diff(M ) such that every p ∈ P (g) (∀g ∈ U(f )) is hyperbolic.Then such a set was characterized as the set of all diffeomorphisms satisfying Axiom A with no-cycles.Theorem 1 will be proved by using the following proposition.This result is already stated in [8, Lemma 3] for int E(M ), and its proof is almost the same as that of [2, Proof of Theorem 1].Here we shall give a rather simple proof for the completeness.
To get the conclusion, it is enough to show that every p ∈ P (f ) is hyperbolic.
Fix a neighborhood U(f ) ⊂ int CE(M ) of f , and by assuming that there is a non-hyperbolic periodic point p = f n (p), we shall derive a contradiction.Here n > 0 is the prime period of p.The tangent space T p M splits into the direct sum p where E u p , E s p and E c p are D p f n -invariant subspaces corresponding to the absolute values of the eigenvalues of D p f n with greater than one, less than one and equal to one, and suppose E c p = 0.Then, for every ε > 0 there exists a linear automorphism O : Then there exists m > 0 and v ∈ E c p \ {0 p } such that v = 1 and G m (tv) = tv (t ≥ 0).For a sufficently small 0 < δ 1 < δ 0 , we have This is a contradiction since exp p v δ1 is a continuum.
To prove Theorem 1 we shall prepair some notations.Let f ∈ Diff(M ) satisfy Axiom A. Thus the non-wandering set of f , Ω(f ), is hyperbolic.For any ε > 0 and for x ∈ Ω(f ), the local stable manifold and the local unstable manifold are denoted by W s ε (x) and W u ε (x) respectively.We may assume that there are ε 0 > 0 and 0 < λ < 1 such that for x ∈ Ω(f ) Proof of Theorem 1: Let f ∈ int CE(M ).Then f satisfy Axiom A by the proposition.We shall show that for every This equality will be proved by using a standard perturbation procedure.
For f ∈ int CE(M ), by assuming that there is Then g ∈ U(f ) so that g is continuum-wise expansive with constant e > 0. However, by the hyperbolicity, it is easy to see that there is 0 < δ < δ 0 /4 such that diam g n (exp x E δ (x)) < e for n ∈ Z.This is a contradiction and so we have f ∈ QA(M ).
To prove Theorem 2 we shall use the following two lemmas.
Then there is a constant c > 0 and a Proof of Theorem 2: The equivalence of (1), ( 2) and (3) follows from [9, Corollary 1] and [13,Theorem].To get the conclusion it is enough to show an implication (5) → (2) because (2) → (4) → (5) is well known.Hereafter let f ∈ QA(M ) has a persistency, and let c > 0 and U(f ) be as in Lemma 1. Fix 0 < ε < c/2 and let U ε (f ) be a C 0 neighborhood of f as in the definition of a persistency.
For any g Thus y = y by Lemma 1 and which is a contradiction.
We denote such y by h g (x) for x ∈ M .Then h ), g n (g(h g (x)))) < ε for n ∈ Z.
Thus, for n ∈ Z From this h • f = g • h is concluded.Clearly we have d(h g (x), x) < ε for x ∈ M .
To prove the continuity of h g , for every α > 0, let N = N (g, α) > 0 be as in Lemma 2. Since f is uniformly continuous, there is β > 0 such that d(x, y) < β (x, y ∈ M ) implies d(f n (x), f n (y)) ≤ c−2ε for −N ≤ n ≤ N .Thus d(g n • h g (x), g n • h g (y)) ≤ d(g n • h g (x), f n (x)) + d(f n (x), f n (y)) + d(f n (y), g n • h g (y)) < 2ε + d(f n (x), f n (y)) ≤ c for −N ≤ n ≤ N .Hence we have d(h g (x), h g (y)) < α. Since + d(g n • h g (y), f n (y)) < 2ε + d(g n • h g (x), g n • h g (y)), if we assume that h g (x) = h g (y), then d(f n (x), f n (y)) < c for all n ∈ Z. Thus x = y and so f is structurally stable.The proof of Theorem 2 is completed.
u and c, all eigenvalues of O • D p f n |E c p are of a root of unity, where I : T p M → T p M is an identity map.By making use of Franks's Lemma (see [2, Lemma 1.1]), we can find δ 0 > 0 and g ∈ U(f ) such that