Average-shadowing Property Research Articles

Shadowing properties and chaotic properties of non-autonomous product systems

<abstract><p>This paper examines how properties such as shadowing properties, transitivity, and accessibility in non-autonomous discrete dynamical systems carry over to their product systems. The paper establishes a proof that the product system exhibits the pseudo-orbit shadowing property (PSP) if, and only if, both factor systems possess PSP. This relationship, which is both sufficient and necessary, also holds for the average shadowing property (ASP) and accessibility. Consequently, in practical problem scenarios, certain chaotic properties of two-dimensional systems can be simplified to those observed in one-dimensional systems. However, it should be noted that while the point-transitivity, transitivity, or mixing of the product system can be deduced from the factor systems, the reverse is not true. In particular, this paper constructs counterexamples to demonstrate that some of the theorems presented herein do not hold when considering their inverses.</p></abstract>

Some Chaotic Properties of G- Average Shadowing Property

Let be a metric space and be a continuous map. The notion of the -average shadowing property ( ASP ) for a continuous map on –space is introduced and the relation between the ASP and average shadowing property(ASP)is investigated. We show that if has ASP, then has ASP for every . We prove that if a map be pseudo-equivariant with dense set of periodic points and has the ASP, then is weakly mixing. We also show that if is a expansive pseudo-equivariant homeomorphism that has the ASP and is topologically mixing, then has a -specification. We obtained that the identity map on has the ASP if and only if the orbit space of is totally disconnected. Finally, we show that if is a pseudo-equivariant map, and the trajectory map is a covering map, then has the ASP if and only if the induced map has ASP.

The average shadowing property and chaos for continuous flows

Let X be a compact metric space and ϕ : R × X → X be a continuous flow. In this paper, we prove that if ϕ has the average shadowing property and the almost periodic points of ϕ are dense in X, then ϕ × ϕ is topologically ergodic. As a corollary, we obtain that if a Lyapunov stable flow ϕ has the average-shadowing property, then X is a singleton.

H–Average-Shadowing Property and Chaos

Let X be a compact metric space and f : X → X be a homeomorphism map. In this paper we are going to introduce the h-average shadowing property for discrete dynamical system (X, f). It is shown that f has the h–average shadowing property if and only if f −1 has the h–average shadowing property. We also prove that if f has the h-average shadowing property, then every point x is chain recurrent. The relation between chaos and shadowing property in discrete dynamical system is considered.

The average-shadowing property and strong ergodicity

Let X be a compact metric space and f:X→X be a continuous map. In this paper, we prove that if f has the average-shadowing property and the minimal points of f are dense in X, then f is weakly mixing and totally strongly ergodic. As applications we obtain that if f is a distal or Lyapunov stable map having the average-shadowing property, then X is consisting of one point. Moreover, we illustrate that the full shift has the average-shadowing property.

The asymptotic average-shadowing property and transitivity for flows

The asymptotic average-shadowing property is introduced for flows and the relationships between this property and transitivity for flows are investigated. It is shown that a flow on a compact metric space is chain transitive if it has positively (or negatively) asymptotic average-shadowing property and a positively (resp. negatively) Lyapunov stable flow is positively (resp. negatively) topologically transitive provided it has positively (resp. negatively) asymptotic average-shadowing property. Furthermore, two conditions for which a flow is a minimal flow are obtained.

The average-shadowing property and topological ergodicity

In this note we prove that a Lyapunov stable map having the average-shadowing property from a compact metric space onto itself is topologically ergodic, but it is not topologically weakly mixing.

Erratum to “The average-shadowing property and transitivity for continuous flows” [Chaos, Solitons and Fractals 23 (2005) 989–995

Erratum to “The average-shadowing property and transitivity for continuous flows” [Chaos, Solitons and Fractals 23 (2005) 989–995

The average-shadowing property and transitivity for continuous flows☆

The average-shadowing property and transitivity for continuous flows☆

The average-shadowing property and topological ergodicity for flows

Abstract In this paper, the transitive property for a flow without sensitive dependence on initial conditions is studied and it is shown that a Lyapunov stable flow with the average-shadowing property on a compact metric space is topologically ergodic.

The average-shadowing property and transitivity for continuous flows

Abstract We prove that a continuous flow with the average-shadowing property is chain transitive and a Lyapunov stable continuous flow with the average-shadowing property is a minimal flow.

Diffeomorphisms with the Average-Shadowing Property on Two-Dimensional Closed Manifolds

The average pseudo-orbits and the averageshadowing property of diffeomorphisms on two-dimensional closed manifolds are considered, and the C1 interior of the set of all diffeomorphisms satisfying the average-shadowing property is characterized as the set of all Anosov diffeomorphisms. The notion of pseudo-orbits very often appears in several branches of the modern theory of dynamical systems, and, especially, the pseudoorbit shadowing property usually plays an important role in the investigation of the stability theory. In [1] Blank introduced the notion of average pseudo-orbits as a certain generalization of the notion of pseudo-orbits (see also [2, p. 19]) and it was proved there that, for a certain kind of hyperbolic system f , every average pseudo-orbit of f is shadowed in average by some true orbit of f (the average-shadowing property). Let M be a C∞ closed manifold, that is, M is compact connected and ∂M = ∅, and let d be the distance induced from a Riemannian metric ‖ · ‖ on TM . Denote by Diff (M) the set of all diffeomorphisms on M endowed with C topology. For δ > 0, a sequence {xi}i=−∞ of points in M is called a δ-average pseudo-orbit of f ∈ Diff (M) if there is a number N = N(δ) > 0 such that for all n ≥ N , k ∈ Z,