Inverse shadowing was introduced by Corless and Pilyugin [1] and also as a part of the concept of bishadowing by Diamond et al. [2]. Kloeden, Ombach and Pokroskii [3] defined this property using the concept of -method. One can also see [4-7] for more information about the concept of -method. Authors in [8] studied on locally genericity of weak inverse shadowing with respect to class. For flows, there are lots of existing work on finding the minimal sets in a systems with shadowing property. See for example [9-12]. In this paper we study diffeomorphisms with weak inverse shadowing property with respect to class as and. First we show that an -stable diffeomorphism has weak inverse shadowing property with respect to classes of continuous method and (Theorem 1) and some -stable diffeomorphisms have weak inverse shadowing property with respect to classes (Theorem 2). In addition we study relation between minimality and weak inverse shadowing property with respect to class and show that a chain transitive homeomorphism on compact metric space is minimal if and only if it has weak inverse shadowing property with respect to class (Theorem 3). Finally we study relation between positively expansive and inverse shadowing property with respect to class and show that if has inverse shadowing property with respect to class, then is not positive expansive (Theorem 4).

Let be a compact metric space and let be a homeomorphism (a discrete dynamical system on). A sequence is called an orbit of, denote by, if for each, and is called a -pseudo-orbit of if

Denote the set of all homeomorphisms of by. In consider the complete metric

which generates the -topology.

Let be the space of all two sided sequence with elements, endowed with the product topology. For let denote the set of all -pseudo orbits of.

A mapping is said to be a -method for if, where is the 0-component of. If is a -method which is continuous then it is called a continuous -method. The set of all -methods (resp. continuous -methods) for will be denoted by (resp.). If is a homeomorphism with, then induces a continuous -method for defined by

Let denote the set of all continuous - methods for which are induced by with.

Let and, a homeomorphism is said to have the inverse shadowing property with respect to the class, , c, h, in if for any there is such that for any -method in and any point there exists a point for which

A homeomorphiosm is said to have weak inverse shadowing property with respect to the class, , c, h, in if for any there is such that for any -method in and any point there exists a point for which

Fix. A continuous -method of class for the diffeomorphism is a sequence, where any is a continuous mapping such that

A sequence is a pseudo-orbit generated by a continuous -method of a class if

Fix. A continuous -method of class for the diffeomophism is a sequence, with for and such that any is a continuous mapping with the property

A sequence is a pseudo-orbit generated by a continuous -method of class if

If a sequence is generated by or we briefly write.

A diffeomorphism is said to have (weak) inverse shadowing property if for any and there exists such that, for any continuous -method, we can find a pseudo-orbit satisfying the inequalities

Pilyugin [5] showed that a structurally stable diffeomoriphism has the inverse shadowing property with respect to classes of continuous method, and. He also showed that any diffeomorphism belonging to the -interior of the set of diffeomorphisms having the inverse shadowing property with respect to classes of continuous method, and is structurally stable.

In this section we show that an -stable diffeomorphism has the weak inverse shadowing property with respect to classes of continuous methods and and if we impose some condition on an -stable diffeomorphism, then it has weak inverse shadowing property with respect to classes.

Theorem 1 If a diffeomorphism is -stable, then it has the weak inverse shadowing property with respect to both classes and.

Before proving this main result, let us briefly recall some definitions. A diffeomorphism is called -stable if there is a -neighborhood of

such that for any, is topologically conjugate to.

A diffeomorfphism is called an Axiom system if is hyperbolic and if.

Axiom and no-cycle systems are -stable [13].

Let be an Axiom diffeomorphism of. By the Smale spectral Decomposition Theorem, the nonwandering set an e represented as a finite union of basic sets.

In the proof of theorem 1 in [5], Pilyugin has used the following statement.

If a -diffeomorphism satisfies Axiom and the strong transversality condition, then there exist constants and and linear subspace, of for such that

and

if and are the projectors in onto parallel to and onto parallel to, respectively, then

(here is the operator norm).

Conditions (1), (2) and (3) play a basic role in the proof of theorem 1 in [5]. If is a basic set then we can see for every, conditions (1), (2) hold. Since is bounded for, standard reasening shows (see, for example, Lemma 12.1 in [14]) that there exists a constant for which inequalities (3) hold. Hence similar to the proof of theorem 1 in [5], has the inverse shadowing property with respect to classes and on. The following two propositions are well known (proposition 1 is the classical Birkhoff theorem [13], for proofs of statements similar to proposition 2, see [15], for example).

proposition 1 Let be a homeomorphism of a compact topological space and be a neighborhood of its nonwandering set. Then there exists a positive integer such that

for every, where is the cardinality of a set.

In the following proposition, we assume that is an -stable diffeomorphism of a closed smooth manifold.

proposition 2 If is a basic set, then for any neighborhood of there exists neighborhood with the following property: if for some and, , then for.

Lemma 1 Let be an -stable diffeomorphism and be the Smale Spectral Decomposition. Let be a neighborhood of for. Then for any there exists and for some, such that

and similarly there exists

Proof. Suppose that the lemma is not true for some. Let be a neighborhood of as in proposition 2. Proposition 1 shows that there exists such that for some. By assumption there exists such that. By proposition 2, for. Thus using proposition 1, there exists such that for some and there exists such that. By proposition 2, for. This process show that

contradicting proposition 1. Proof of

is similar.

Proof of theorem 1. Let and be arbitrary. Let be a neighborhood of for, such that shadowing property hold for them. By lemma 1 there exists a positive number, such that

for some. Since is compact, there exist, such that

where is as in the shadowing theorem for hyperbolic set. So

is a periodic -pseudo-orbit of in. By shadowing theorem for hyperbolic sets, there iswhich -shadows. This shows that

But has the inverse shadowing property with respect to classes and. Thus there exists such that for any continuous -method, we can find a pseudo-orbit satisfying

Inequalities and show that

. This complete the proof of theorem 1.

Theorem 2 Let be an -stable diffeomorphism and be the Smale Spectral Decomposition such that be fix point sources or sinks. Then has the weak inverse shadowing property with respect to class in, where is set of fix points of.

Proof. Let and be arbitrary that is not fix point and be open neighborhoods of respectively with diameter less than. Lemma 1 shows that there exists and and for some, such that

and

Note that is a neighborhood of fix point sink and is a neighborhood of fix point source. Choose

such that

for every with, where

This shows that if is a -pseudo orbit and

then. there exists

such that

and

Choose such that if

then

And also for.

So for any -pseudo orbit

we have

Now for any ()-method, by regarding the process of choosing and (4), (5), (6) we have, and this completes the proof of theorem 2.

The following example shows that an -stable maybe has not the weak inverse shadowing property with respect to class in its fix point.

Example. Represent as the sqare, with identified opposite sides. Let be a diffeomorphism with the following properties:

The nonwandering set of is the union of 4 hyperbolic fixed points, that is, , where is a source, is a sink, and are saddles;

where and are the stable and unstable manifolds, respectively.

There exist neighborhoods of such that for

The eigenvalues of are with , and the eigenvalues of are with.

Plamenevskaya [16] showed that has the weak shadowing property if and only if the number is irrational. Note that does not have the shadowing property. We can see that does not have the weak inverse shadowing property with respect to class as well (Note that the number is not necessary irrational). For any, let be the number of the weak inverse shadowing property of. Construct a -method as following:

where and.

For every, define

A homeomorphism is called minimal if, A closed, implies either or. It is easy to see that is minimal if and only if for every.

A homeomorphiosm is said to be chain transitive if for every and there are -pseudoorbits from to and from to.

The following example shows that there exists homeomorphiosms with inverse shadowing property with respect to class which is not minimal.

Example. Let with metric

Let be a permutation of the set for some. Let if, and otherwise.

is a homeomorphism and every point of is a periodic point for. We claim has weak inverse shadowing property with respect to class.

  Proof of claim. Given choose such that

. if and only if where. Let and be a -method.

Let, then implies for and hence by definition of, for. Also implies for.

Hence if and then

for, and so.

Using this procedure we will get

for. A similar reasoning with having in mind that is a homeomorphism proves that

for. Hence for and

has inverse shadowing property with respect to. It is easy to see that is not minimal.

Theorem 3 Let be a chain transitive homeomorphism on compact metric space. Then is minimal if and only if has weak inverse shadowing property with respect to class.

Proof. Suppose that has weak inverse shadowing property with respect to class and. Let be an open set in. Choose and such that. There is such that for each -method, there is such that

For every, there is a -chain,

from to. Consider

as a -pseudo-orbit, such that it’s 0-component be. Construct a -method such that.

Hence there is such that, and so for some. Therefore . This shows that each orbit of is dense in and so is minimal. The converse i.e. to see that each minimal homeomorphism has weak inverse shadowing property with respect to class, is obvious.

A homeomorphism on metric space is said expansive if there exists constant such that for every there exists integer number

such that.

Theorem 4 If homeomorphism on metric space has the inverse shadowing property with respect to class, then is not expansive.

Proof. Suppose that is expansive and has the inverse shadowing property with respect to class. Let be as in definition of expansivity and be such that for any -method in and any point there exists a point for which

Let be arbitrary. Choose such that and. Construct a -method as following.

For any define

and

Since has the inverse shadowing property with respect to class, for there exists such that

By regarding to choose of -method, we have

for some, that contradicts the expansivity of. This completes the proof of theorem.

In this paper we showed that an -stable diffeomorphism has the weak inverse shadowing property with respect to classes of continuous method and and some of the -stable diffeomorphisms have weak inverse shadowing property with respect to classes. In addition we studied relation between minimality and weak inverse shadowing property with respect to class and relation between expansivity and inverse shadowing property with respect to class.