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In this article, we introduce the notion of almost average shadowing property (ALASP) for continuous semi-flows and study its properties. We give a class of continuous semi-flows possessing the ALASP. We also investigate the relationship of the ALASP with various known dynamical properties for continuous semi-flows, for instance, topological transitivity, weak mixing, strong ergodicity, equicontinuity, sensitivity, Li-Yorke chaos and Takens and Ruelle chaos. [ABSTRACT FROM AUTHOR]

The aim of this paper is to relate the almost average shadowing property (ALASP) with various variants of ergodicity and with some other dynamical properties. We also study the relation of the ALASP with proximality. [ABSTRACT FROM AUTHOR]

In the present paper we study the C 1 -robustness of the three properties: average shadowing, asymptotic average shadowing and limit shadowing within two classes of conservative flows: the incompressible and the Hamiltonian ones. We obtain that the first two properties guarantee dominated splitting (or partial hyperbolicity) on the whole manifold, and the third one implies that the flow is Anosov. [ABSTRACT FROM AUTHOR]

Abstract: Let X be a compact metric space and f: X → X be a continuous map. In this paper, we investigate the relationships between the asymptotic average shadowing property (Abbrev. AASP) and other notions known from topological dynamics. We prove that if f has the AASP and the minimal points of f are dense in X, then for any n ⩾1, f × f ×⋯× f(n times) is totally strongly ergodic. As a corollary, it is shown that if f is surjective and equicontinuous, then f does not have the AASP. Moreover we prove that if f is point distal, then f does not have the AASP. For f: [0,1]→[0,1] being surjective continuous, it is obtained that if f has two periodic points and the AASP, then f is Li–Yorke chaotic. [Copyright &y& Elsevier]

Abstract: In this paper, we present a novel approach to encrypt a message (a text composed by some alphabets) using chaos and shadowing. First, we generate a numerical chaotic orbit based on the logistic map, and use the shadowing algorithm of Smaoui and Kostelich [Smaoui N, Kostelich E. Using chaos to shadow the quadratic map for all time. Int J Comput Math 1998;70:117–29] to show that there exists a finite number of true orbits that shadow the numerical orbit. Then, the finite number of maps generated is used in Baptista’s algorithm [Baptista MS. Cryptography with chaos. Phys Lett A 1998;240:50-4] to encrypt each character of the message. It is shown that the use of chaos and shadowing in the encryption process enhances the security level. [Copyright &y& Elsevier]

Abstract: Let X be a divergence-free vector field on a three-dimensional compact connected Riemannian manifold. In this paper, we show that if X is in the C 1-interior of the set of divergence-free vector fields which satisfy the average shadowing property then X is Anosov. We also obtain similar result for asymptotic average shadowing property. [Copyright &y& Elsevier]