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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

Let M be a closed smooth Riemannian manifold and let f : M → M be a diffeomorphism. We show that if f has the C1 robustly asymptotic orbital shadowing property then it is an Anosov diffeomorphism. Moreover, for a C1 generic diffeomorphism f, if f has the asymptotic orbital shadowing property then it is a transitive Anosov diffeomorphism. In particular, we apply our results to volume-preserving diffeomorphisms.

In this paper, we discuss the two-sided limit shadowing property for continuous flows defined in compact metric spaces. We analyse some of the results known for the case of homeomorphisms in the case of continuous flows and observe that some differences appear in this scenario. We prove that the suspension flow of a homeomorphism satisfying the two-sided limit shadowing property also satisfies it. This gives a lot of examples of flows satisfying this property, however, it enlighten an important difference between the case of flows and homeomorphisms. There are flows satisfying the two-sided limit shadowing property that are not topologically mixing, while homeomorphisms satisfying the two-sided limit shadowing property satisfy even the specification property. It can happen that a suspension flow has the two-sided limit shadowing property but the base homeomorphism does not, though it is proved that it must satisfy a strictly weaker property called two-sided limit shadowing with a gap (as in [B. Carvalho, D. Kwietniak. On homeomorphisms with the two-sided limit shadowing property, J. Math Anal. Appl.420 (2014), 801–813]). We define a similar notion of two-sided limit shadowing with a gap for flows and prove that these notions are actually equivalent in the case of flows. Finally, we prove that singular suspension flows (in the sense of [Komuro: One-parameter flows with the pseudo orbit tracing property. Mh. Math.98 (1984) 219–253]) do not satisfy the two-sided limit shadowing property. [ABSTRACT FROM AUTHOR]

In this paper, we will prove some fixed point results using a Pompeiu type metric on Pcl(X), for multi-valued operators satisfying two conditions: contractivity and monotonicity. The approach is based on a fixed point theorem for a multi-valued operator in the setting of a b-metric space. Several qualitative properties(well-posedness, Ulam-Hyers stability) are also obtained. In the second part of this paper, we will consider the extended coupled fixed point problem for a multi-valued operator. We will consider the problem of the existence of the solutions. Data dependence and well-posedness of the extended coupled fixed point problem are also discussed. [ABSTRACT FROM AUTHOR]

We prove that the set of maps which exhibit distributional chaos of type 1 (DC1) is C0-dense in the space of continuous self-maps of given any compact topological manifold (possibly with boundary). [ABSTRACT FROM AUTHOR]

In this paper, we introduce the shadowing, limit shadowing and the shadow by two- sided limit. These properties are related to each other according to specifice terms. We see that in the sence of positive expansive map the h–shadowing is equivalent to strong and continuous shadowing, and L–shadowing is also equivalent to strong and continuous shadowing. Also we obtain the relation among L–shadowing, h–shadowing and s–limit shadowig. Finly, we find the relation among : shadowing, h–shadowing, L–shadowing, limit shadowing, s–limit shadowig, strong shadowing, continuous shadowing, the shadowing by two- sided limit, and the shadowing by two - sided limit with gap, they are equivalent in sence of transitive positive expancive map on compact spaces. [ABSTRACT FROM AUTHOR]

We discuss further the dynamics of n-expansive homeomorphisms with the shadowing property, started in Carvalho and Cordeiro (J Differ Equ 261:3734–3755, 2016). The L-shadowing property is defined and the dynamics of n-expansive homeomorphisms with such property is explored. In particular, we prove that positively n-expansive homeomorphisms with the L-shadowing property can only be defined in finite metric spaces. [ABSTRACT FROM AUTHOR]

Let be a symplectic diffeomorphism on a closed -dimensional Riemannian manifold . In this paper, we show that is Anosov if any of the following statements holds: belongs to the -interior of the set of symplectic diffeomorphisms satisfying the limit shadowing property or belongs to the -interior of the set of symplectic diffeomorphisms satisfying the limit weak shadowing property or belongs to the -interior of the set of symplectic diffeomorphisms satisfying the s-limit shadowing property. [ABSTRACT FROM AUTHOR]

We discuss the dynamics of n -expansive homeomorphisms with the shadowing property defined on compact metric spaces. For every n ∈ N , we exhibit an n -expansive homeomorphism, which is not ( n − 1 ) -expansive, has the shadowing property and admits an infinite number of chain-recurrent classes. We discuss some properties of the local stable (unstable) sets of n -expansive homeomorphisms with the shadowing property and use them to prove that some types of the limit shadowing property are present. This deals some direction to the problem of non-existence of topologically mixing n -expansive homeomorphisms that are not expansive. [ABSTRACT FROM AUTHOR]

In this paper, we will consider coupled fixed point problems in b-metric spaces for single-valued operators satisfying a symmetric contraction condition. On one hand, existence and uniqueness of the solution and, on the other hand, data dependence, well-posedness, Ulam-Hyers stability, limit shadowing property of the coupled fixed point problem are discussed. The approach is based on the application of a Ran-Reurings type fixed point theorem for an appropriate operator on the Cartesian product space. Some applications to a system of integral equations and to a periodic boundary value problems are also given. [ABSTRACT FROM AUTHOR]

Let be a compact metric space, and let be a homeomorphism. We show that if has the limit shadowing property then is chaotic in the sense of Li-Yorke. Moreover, is dense Li-Yorke chaos. [ABSTRACT FROM AUTHOR]

In this paper, we will consider the coupled fixed point problem for single-valued operators satisfying a symmetric contraction condition with respect to maximum. An application to a periodic boundary value problem illustrates the results. The study of a coupled coincidence problem is also suggested. [ABSTRACT FROM AUTHOR]

We present the following: (1) if a volume-preserving diffeomorphism has C1-robustly various limit shadowing property, then it is Anosov; (2) C1-generically, if a volume-preserving diffeomorphism has various limit shadowing property, then it is Anosov. [ABSTRACT FROM AUTHOR]

We explore the notion of the two-sided limit shadowing property introduced by Pilyugin in 2007. Indeed, we characterize the C 1-interior of the set of diffeomorphisms with such a property on closed manifolds as the set of transitive Anosov diffeomorphisms. As a consequence we obtain that all codimension-one Anosov diffeomorphisms have the two-sided limit shadowing property. We also prove that every diffeomorphism f with such a property has neither sinks nor sources and is transitive Anosov (in the Axiom A case). In particular, no Morse-Smale diffeomorphism has the two-sided limit shad-owing property. Finally, we prove that C¹-generic diffeomorphisms with the two-sided limit shadowing property are transitive Anosov. All these results allow us to reduce the well-known conjecture about the transitivity of Anosov diffeomorphisms to prove that the set of diffeomorphisms with the two-sided limit shadowing property coincides with the set of Anosov diffeomorphisms. [ABSTRACT FROM AUTHOR]

We prove that the two-sided limit shadowing property is among the strongest known notions of pseudo-orbit tracing. It implies shadowing, average shadowing, asymptotic average shadowing and specification properties. We also introduce a weaker notion that is called two-sided limit shadowing with a gap and prove that it implies shadowing and transitivity. We show that those two properties allow to characterize topological transitivity and mixing in a class of expansive homeomorphisms and hence they characterize transitive (mixing) shifts of finite type. [ABSTRACT FROM AUTHOR]

It is known that if a closed invariant set Λ of a diffeomorphism f of a compact smooth manifold M is hyperbolic then f has the pseudo orbit tracing property. For a weak notion of hyperbolicity, in [4] if a transitive set of a diffeomorphism f of the three dimensional manifold M has a partially hyperbolic structure then f does not have the pseudo orbit tracing property. From the results, we consider a compact smooth manifold M which the dimension is greater than three. It is a general version of the three dimensional manifold M. More detail, we show that if a transitive diffeomorphism f of a compact smooth manifold M is partially hyperbolic, then f does not have the pseudo orbit tracing property. Moreover, if f is partially hyperbolic on M then f does not have the asymtotic and the ergodic pseudo orbit tracing properties. [ABSTRACT FROM AUTHOR]

This paper introduces the notion of a general approximation property, which encompasses many existing types of shadowing. It is proven that there exists a metric space X such that the sets of maps with many types of general approximation properties (including the classic shadowing, the L-shadowing, limit shadowing, and the s-limit shadowing) are not dense in C( X), S( X), and H( X) (the space of continuous self-maps of X, continuous surjections of X onto itself, and self-homeomorphisms of X) and that there exists a manifold M such that the sets of maps with general approximation properties of nonlocal type (including the average shadowing property and the asymptotic average shadowing property) are not dense in C( M), S( M), and H( M). Furthermore, it is proven that the sets of maps with a wide range of general approximation properties (including the classic shadowing, the L-shadowing, and the s-limit shadowing) are dense in the space of continuous self-maps of the Cantor set. A condition is given that guarantees transfer of general approximation property from a map on X to the map induced by it on the hyperspace of X. It is also proven that the transfer in the opposite direction always takes place. [ABSTRACT FROM AUTHOR]

In this paper, various shadowing properties are considered for a positively expansive map on a compact metrizable space. We show that the Lipschitz shadowing property, the s-limit shadowing property and the strong shadowing property are all equivalent to the (usual) shadowing property for a positively expansive map. Furthermore, for a positively expansive open map, the average shadowing property is shown. [Copyright &y& Elsevier]

Abstract The purpose of this paper is to present a fixed point theory for multivalued φ-contractions using the following concepts: fixed points, strict fixed points, periodic points, strict periodic points, multivalued Picard and weakly Picard operators; data dependence of the fixed point set, sequence of multivalued operators and fixed points, Ulam-Hyers stability of a multivalued fixed point equation, well-posedness of the fixed point problem, limit shadowing property of a multivalued operator, set-to-set operatorial equations and fractal operators. Our results generalize some recent theorems given in Petruşel and Rus (The theory of a metric fixed point theorem for multivalued operators, Proc. Ninth International Conference on Fixed Point Theory and its Applications, Changhua, Taiwan, July 16-22, 2009, 161-175, 2010). 2010 Mathematics Subject Classification 47H10; 54H25; 47H04; 47H14; 37C50; 37C70

In this paper, some existence results for a system of operator inclusions are presented. Qualitative properties of the solution set are also discussed. The method is based on the application of a fixed point theorem for an appropriate operator on the Cartesian product of the given spaces. The approach is new even for the case of the metric spaces. As an application, an existence result for a mixed boundary and initial value problem for a system of second order differential inclusions is given. [ABSTRACT FROM AUTHOR]