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Introduction The influence of persistence behavior of a dynamical system on tangent bundle of a manifold is always a challenge in dynamical systems. Persistence properties have been studied on whole manifold or on some pieces with independent dynamics. Since shadowing property has an important role in the qualitative theory of dynamical systems, by focusing on various shadowing properties, such as usual shadowing, inverse shadowing, limit shadowing, many interesting results have been obtained. The notion of limit shadowing property introduced by S. Pilyugin who obtained its relation to other various shadowing. Blank introduced the notion of average-shadowing property. It is known that every Axiom A diffeomorphism restricted to a basic set has the average shadowing property. K. Sakai proved that the -interior of the set of all diffeomorphisms satisfying the average-shadowing property is characterized as the set of all Anosov diffeomorphisms. Asymptotic average shadowing (AASP) defined by R. Gu for continuous maps, combines to the limit shadowing property with the average shadowing property. Here we modify the notion (AASP) and define the limit average shadowing for diffeomorphisms (LASP). R. Gu presented some basic properties of the limit average shadowing for continuous maps. He proved that if a continuous map has the limit average shadowing on a compact metric space, then is chain transitive and that -hyperbolic homeomorphisms with limit average shadowing are topologically transitive. M. Kulczycki et.al., found some relations between LASP and the other notion of topological dynamics. They proved that a surjective map with specification property has the LASP. Also, they found the relation between LASP and shadowing property. They also have been shown that an expansive continuous map with shadowing property is LASP if and only if it is mixing. This paper follows the ideas of R. Gu and M. Kulczycki et.al. Here we define LASP for diffeomorphism with a slight modification of the continuous case. We give an example which shows that shadowing property and LASP are not equivalent. Also, we introduce the notion of - stably limit average shadowing for a closed -invariant subset of and show that if is -stably limit average shadowing and the minimal points of are dense there, then admits a dominated splitting. Statement of the results In this paper we give a system which has the limit average shadowing, but not the shadowing property. Also, one can give examples which have the shadowing but not the limit average shadowing property. Thus the limit average shadowing property does not imply the shadowing property. In fact, we can give a class of diffeomorphisms which have LASP, but not the shadowing property. In fact the following proposition gives a large class of diffeomorphisms satisfying the limit average shadowing. Proposition A. Let be a locally maximal -invariant set. If is the specific set for then is limit average shadowable. The main purpose of the paper is to characterize the closed -invariant set via limit average shadowing property in -open condition. So, we consider the notion of limit average shadowing property in geometric differential dynamical systems. First we show that if has the limit average shadowing property on a closed -invariant set then is chain transitive. By using chain transitivity and limit average shadowing property we can prove that is transitive. Proposition B. If is -stably limit average shadowable, then there is a neighborhood of and a neighborhood of such that contains neither almost sinks nor almost sources for any Since we have proved that if has the limit average shadowing property on a closed f-invariant set and minimal points of are dense then is transitive. It is essentially proved that under assumptions and conclusions of the Proposition B, admits a dominated splitting. Thus we get the main result of this paper. Theorem C. Let be a closed f-invariant set whose minimal points are dense there. If is -stably limit average shadowing then admits a dominated splitting../files/site1/files/62/2Abstract.pdf

The multiple flow systems (briefly; MFS) and orbits on this systems are introduced, actually in a multiple flow system we have some flows instead of some functions in iterated function systems. Then the asymptotic average shadowing property, chain transitivity and attractor sets are generalized to these new type of systems. We consider uniformly contracting and uniformly expanding multiple flow systems and show that every uniformly contracting, or uniformly expanding multiple flow system has the asymptotic shadowing property. We also prove that if a MFS, F on a compact metric space X has the asymptotic average shadowing property, then F is chain transitive, but the converse is not always true. As a result, this prove that if F is an uniformly contracting or uniformly expanding MFS on compact metric space X, then X is the only nonempty attractor of F . [ABSTRACT FROM AUTHOR]

This paper investigates the average shadowing property and the asymptotic average shadowing property of linear dynamical systems in Banach spaces. Firstly, necessary and sufficient conditions for an invertible operator T on a Banach space to have the average shadowing property and the asymptotic average shadowing property are given, respectively. Then, it is concluded that both the average shadowing property and the asymptotic average shadowing property are preserved under iterations. Furthermore, if T is hyperbolic, then T has the (asymptotic) average shadowing property. However, the inverse implication fails in infinite-dimensional Banach spaces. Finally, it is proved that the (asymptotic) average shadowing property is equivalent to the hyperbolicity for dynamical systems in a finite-dimensional Banach space. [ABSTRACT FROM AUTHOR]

This paper proves that the linear transformation 𝒜 (x) = A x on ℂ n has the (asymptotic) average shadowing property if and only if A is hyperbolic, where A is a nonsingular matrix, giving a positive answer to a question in [Lee, 2012a]. Besides, it is proved that 𝒜 does not have the d -shadowing property, thus does not have the ergodic shadowing property for every nonsingular matrix A. [ABSTRACT FROM AUTHOR]

We show that if a C¹ generic diffeomorphism of a closed smooth two-dimensional manifold has the average shadowing property or the asymptotic average shadowing property, then it is Anosov. Moreover, if a C¹ generic vector field of a closed smooth three-dimensional manifold has the average shadowing property or the asymptotic average shadowing property, then it satisfies singular Axiom A without cycles. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the notion of the asymptotic average shadowing property to parameterized IFS's and prove some related theorems on this notion. Specially, it is proved that every uniformly contracting IFS has the asymptotic average shadowing property. As an important result, we show that if a continuous surjective IFS $$\mathcal {F}$$ on a compact metric space has the asymptotic average shadowing property then $$\mathcal {F}$$ is chain transitive. Moreover, we give some examples to illustrate our approach and compare the asymptotic average shadowing property for IFS with the asymptotic average shadowing property in discrete dynamical systems. For example, we will show that there is an IFS $$\mathcal {F}$$ which has the asymptotic average shadowing property but does not satisfy the shadowing property. [ABSTRACT FROM AUTHOR]

We show that if a C 1 generic diffeomorphism of a closed smooth two-dimensional manifold has the average shadowing property or the asymptotic average shadowing property, then it is Anosov. Moreover, if a C 1 generic vector field of a closed smooth three-dimensional manifold has the average shadowing property or the asymptotic average shadowing property, then it satisfies singular Axiom A without cycles.

In this paper, we explore a topological system f : M → M with asymptotic average shadowing property and extend Sigmund's results from Bowen's specification case. We show that every non-empty, compact and connected subset V ⊆ M inv ( f ) coincides with some V f ( y ) . Moreover, we show that the set M V = { y ∈ M : V f ( y ) = V } is dense in Δ V = ⋃ ν ∈ V supp ( ν ) ¯ . In particular, if Δ max = ⋃ ν ∈ M inv ( f ) supp ( ν ) ¯ coincides with M , then M max = { y : V f ( y ) = M inv ( f ) } is residual in M . As consequences, we have several corollaries. One is that every invariant measure has generic points. Another is that the set consisting of those points for which the Birkhoff ergodic average does not exist (called irregular set) is either dense in Δ max (residual provided that Δ max = M ) or empty. In particular, we give an uncountable division of irregular set and obtain a refined characterization which can be as a substantial generalization of [18] with a different method. Remark that for systems with asymptotic average shadowing property, this article studies Birkhoff ergodic average from topological viewpoint but it is still unknown how about the theory from the perspective of dimension theory. [ABSTRACT FROM AUTHOR]

Abstract: We prove that if for a continuous map on a compact metric space , the chain recurrent set, has more than one chain component, then does not satisfy the asymptotic average shadowing property. We also show that if a continuous map on a compact metric space has the asymptotic average shadowing property and if is an attractor for , then is the single attractor for and we have . We also study diffeomorphisms with asymptotic average shadowing property and prove that if is a compact manifold which is not finite with , then the interior of the set of all diffeomorphisms with the asymptotic average shadowing property is characterized by the set of -stable diffeomorphisms. [Copyright &y& Elsevier]