Inverse Shadowing and Weak Inverse Shadowing Property

In this paper we show that an  -stable diffeomorphism f has the weak inverse shadowing property with respect to classes of continuous method s  and c  and some of the  -stable diffeomorphisms have weak inverse shadowing property with respect to classes . In addition we study relation between minimality and weak inverse shadowing property with respect to class and relation between expansivity and inverse shadowing property with respect to class . 0 


Introduction
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][5][6][7] for more information about the concept of  -method.Authors in [8] studied on locally genericity of weak inverse shadowing with respect to class 0 .For flows, there are lots of existing work on finding the minimal sets in a systems with shadowing property.See for example [9][10][11][12].In this paper we study diffeomorphisms with weak inverse shadowing property with respect to class as   and 0  .First we show that an -stable diffeomorphism  f has weak inverse shadowing property with respect to classes of continuous method s  and c  (Theorem 1) and some -stable diffeomorphisms have weak inverse shadowing property with respect to classes 0 (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  0   f on compact metric space X is minimal if and only if it has weak inverse shadowing property with respect to class 0  (Theorem 3).Finally we study relation between positively expansive and inverse shadowing property with respect to class 0  an ow that if d sh f has nverse shadowing property with respect to class 0  , t n i Let   , X d be a compact metric space and let : f X X  be a homeomorphism (a discrete dynamical system on X ).A sequence n is called an orbit of Denote the set of all homeomorphisms of X by   Z X .In   Z X consider the complete metric The set of all  -methods (resp.continuous  -methods) for f will be denoted by A homeomorphiosm f is said to have weak inverse shadowing property with respect to the class the diffeomorphism f is a sequence , where any Pilyugin [5] showed that a structurally stable diffeomoriphism has the inverse shadowing property with respect to classes of continuous method, c  and s  .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 if we impose some condition on an  -stable diffeomorphism, then it has weak inverse shadowing property with respect to classes . 0

then it has the weak inverse shadowing property with respect to both classes c
 and s  .
Before proving this main result, let us briefly recall some definitions.A diffeomorphism : Axiom A and no-cycle systems are -stable [13]. Let f be an Axiom A diffeomorphism of M .By the Smale spectral Decomposition Theorem, the nonwandering set   f  i 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 1 C f satisfies Axiom A and the strong transversality condition, then there exist constants and > 0 (here . is the operator norm).di a basic role in the Con tions (1), ( 2) and (3) play proof of theorem 1 in [5].If i  is a basic set then we can see for every   , standard reasening sho (see, for example, Le 12.1 in [14]) that there exists a constant C for which inequalities (3) hold.Hence similar to th proof of theorem 1 in [5], ws mma e f has the inverse shadowing property with respect to classes s  and c  on i  .The following two propositions are well kno n (pr osition 1 is the classical Birkhoff theorem [13], for proofs of statements similar to proposition 2, see [15], for example).
proposition 1 Let w op f be a homeomorphism of a compa e ct topological spac X and U be a neighborhood of its nonwandering set.T n there xists a positive integer N such that for every x X  , where is the cardinality of a Card A set A .
In e that the following proposition, we assum f is an  -stable diffeomorphism of a closed smooth ma fold.

proposition 2 If i
 is a basic set, then for any ni neighborhood U of i there exists neighborhood V with the following property: if for some and similarly there exists j U Proof.Suppose that the lemma is not true for some x M  .Let i V be a neighborhood of i  as in pro-2.Proposition 1 shows that there exists is as in the shadow g theorem for in

Let
. This com lete the proof of theorem 1.
p Theorem 2 f be -stable diffeomorphism d an an Note that is a neighborhood of fix point sink and and and .
. there exists and bit and Choose And also for any  0 1 Now for any ( )-method 0 1 N    , by regard process of choosi and ( 5), (6) we have mple shows that an  -stable maybe has not are the stable and unsta-

on between Minimality an roperty wi 
A ho

Relati to Class
It is easy to see that f is minimal if and only if is said to be chain transitive if for every , x y X  and > 0  there are  -pseudo- x to y and from y to x .orbits from The following example shows that there exists homeof with in morphi s osm ve ich rse shadowing pr with respect t class wh is not minimal.
for some be a per- x y wh e er 0 0 Using this procedure we will get for oning in mind 0 i  .A similar reas with having that n f is a homeom ves that orphism pro as a  -pseu it, such that 's 0-comp do-orb it onent be x .
Const uct a r and so for some Therefore

Conclusion
In


and is called a  -pseudo-orbit of f if contradicts the expansivity of f .This completes the roof of theorem.p t - a homeomorphism f is said to have the inverse shadowing property with respect to the class  sequence is generated by c

2. Diffeomorphisms with Weak Inverse Shadowing Property with Respect to Class θ s , θ c and 0 
s and c and i x otherwise.