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14 pages, 307 KiB

Open AccessFeature PaperArticle

On the Relation between Topological Entropy and Restoration Entropy

by Christoph Kawan

Entropy 2019, 21(1), 7; https://doi.org/10.3390/e21010007 - 23 Dec 2018

Cited by 5 | Viewed by 3348

In the context of state estimation under communication constraints, several notions of dynamical entropy play a fundamental role, among them: topological entropy and restoration entropy. In this paper, we present a theorem that demonstrates that for most dynamical systems, restoration entropy strictly exceeds topological entropy. This implies that robust estimation policies in general require a higher rate of data transmission than non-robust ones. The proof of our theorem is quite short, but uses sophisticated tools from the theory of smooth dynamical systems. Full article

(This article belongs to the Special Issue Entropy in Networked Control)

7 pages, 245 KiB

Open AccessArticle

A Type of the Shadowing Properties for Generic View Points

by Manseob Lee

Axioms 2018, 7(1), 18; https://doi.org/10.3390/axioms7010018 - 20 Mar 2018

Cited by 5 | Viewed by 3391

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Shadowing in Dynamical Systems)

18 pages, 267 KiB

Open AccessReview

Open and Dense Topological Transitivity of Extensions by Non-Compact Fiber of Hyperbolic Systems: A Review

by Viorel Nitica and Andrei Török

Axioms 2015, 4(1), 84-101; https://doi.org/10.3390/axioms4010084 - 4 Feb 2015

Cited by 2 | Viewed by 5300

Abstract

Currently, there is great renewed interest in proving the topological transitivity of various classes of continuous dynamical systems. Even though this is one of the most basic dynamical properties that can be investigated, the tools used by various authors are quite diverse and are strongly related to the class of dynamical systems under consideration. The goal of this review article is to present the state of the art for the class of Hölder extensions of hyperbolic systems with non-compact connected Lie group fiber. The hyperbolic systems we consider are mostly discrete time. In particular, we address the stability and genericity of topological transitivity in large classes of such transformations. The paper lists several open problems and conjectures and tries to place this topic of research in the general context of hyperbolic and topological dynamics. Full article

(This article belongs to the Special Issue Topological Groups: Yesterday, Today, Tomorrow)

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