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Axioms
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Shadowing in Dynamical Systems
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Shadowing in Dynamical Systems

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Geometry and Topology".

Deadline for manuscript submissions: closed (31 May 2019) | Viewed by 18842

Special Issue Editor

Prof. Dr. Kazuhiro Sakai

E-Mail Website
Guest Editor
Department of Mathematics, Utsunomiya University, Utsunomiya 321-8505, Japan
Interests: shadowing property; dynamical systems theory; bifurcation theory; ergodic theory; vector fields; chaos theory
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Since Anosov and Bowen's works, the notion of pseudo-orbits very often appears in several branches of the modern theory of dynamical systems, and, especially, the pseudo-orbits shadowing property (in what follows, this is called the shadowing property) usually plays an important part, not only in the numerical study of dynamical systems, but also in the qualitative study of dynamical systems. In fact, the shadowing property has been applied to the modern theory of structural stability and has played one of the main roles in the global theory of dynamical systems. The shadowing theory of dynamical systems is now an important and rapidly developing branch of the mordern theory of dynamical systems.

Various types of shadowing properties have been introduced in the literature since Anosov and Bowen's works, and, nowadays, these notions are intensively studied by many authors in the platform of topological dynamical systems. Many essential and interesting results have been obtained from the view point of, for instance, measure theory, chaos theory, and combinatorics.

In this Special Issue, by collecting recent achievements on the shadowing property from the dynamical systems community around the world, we would like to spur the study of shadowing theory to explore the new directions and further developments in the theory.

In this issue, we particularly seek contributions on the following three topics:

  • new results on the shadowing property in the frameworks of uniformly hyperbolic systems, non-uniformly hyperbolic systems and topological dynamical systems
  • new results on the shadowing property intertwined with bifurcation theory, ergodic theory, and so on.
  • survey articles which present significant (new or not so new) open questions.

Prof. Dr. Kazuhiro Sakai
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Axioms is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • shadowing property
  • average shadowing property
  • limit shadowing property
  • weak shadowing property
  • orbital shadowing property
  • shadowing measures
  • topologically stable
  • uniformly hyperbolic and non-uniformly hyperbolic
  • dominated splittings
  • singular hyperbolic
  • expansive
  • continuum-wise expansive
  • expansive measures
  • bifurcation theory
  • ergodic theory
  • chaos theory

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Related Special Issue

Published Papers (6 papers)

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Research

6 pages, 273 KiB  
Article
Generic Homeomorphisms with Shadowing of One-Dimensional Continua
by Alfonso Artigue and Gonzalo Cousillas
Axioms 2019, 8(2), 66; https://doi.org/10.3390/axioms8020066 - 26 May 2019
Cited by 1 | Viewed by 2518
Abstract
In this article, we show that there are homeomorphisms of plane continua whose conjugacy class is residual and have the shadowing property. Full article
(This article belongs to the Special Issue Shadowing in Dynamical Systems)
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Figure 1

4 pages, 192 KiB  
Article
A Note on Anosov Homeomorphisms
by Mauricio Achigar
Axioms 2019, 8(2), 54; https://doi.org/10.3390/axioms8020054 - 1 May 2019
Cited by 6 | Viewed by 3297
Abstract
For an α -expansive homeomorphism of a compact space we give an elementary proof of the following well-known result in topological dynamics: A sufficient condition for the homeomorphism to have the shadowing property is that it has the α -shadowing property for one-jump [...] Read more.
For an α -expansive homeomorphism of a compact space we give an elementary proof of the following well-known result in topological dynamics: A sufficient condition for the homeomorphism to have the shadowing property is that it has the α -shadowing property for one-jump pseudo orbits (known as the local product structure property). The proof relies on a reformulation of the property of expansiveness in terms of the pseudo orbits of the system. Full article
(This article belongs to the Special Issue Shadowing in Dynamical Systems)
10 pages, 228 KiB  
Article
Relations between Shadowing and Inverse Shadowing in Dynamical Systems
by Alexey A. Petrov
Axioms 2019, 8(1), 11; https://doi.org/10.3390/axioms8010011 - 17 Jan 2019
Cited by 1 | Viewed by 2798
Abstract
In this paper, we study relations between shadowing and inverse shadowing for homeomorphisms of a compact space. We present an example of a smooth diffeomorphism of a compact three-dimensional manifold that has the shadowing property and does not have the inverse shadowing property. [...] Read more.
In this paper, we study relations between shadowing and inverse shadowing for homeomorphisms of a compact space. We present an example of a smooth diffeomorphism of a compact three-dimensional manifold that has the shadowing property and does not have the inverse shadowing property. For some classes of inverse shadowing, we construct examples of homeomorphisms that have the inverse shadowing property but do not have the shadowing property. Full article
(This article belongs to the Special Issue Shadowing in Dynamical Systems)
10 pages, 239 KiB  
Article
Diffeomorphisms with Shadowable Measures
by Kazumine Moriyasu, Kazuhiro Sakai and Naoya Sumi
Axioms 2018, 7(4), 93; https://doi.org/10.3390/axioms7040093 - 7 Dec 2018
Cited by 5 | Viewed by 3244
Abstract
In this paper, the notion of shadowable measures is introduced as a generalization of the shadowing property from the measure theoretical view point, and the set of diffeomorphisms satisfying the notion is considered. The dynamics of the C 1 -interior of the set [...] Read more.
In this paper, the notion of shadowable measures is introduced as a generalization of the shadowing property from the measure theoretical view point, and the set of diffeomorphisms satisfying the notion is considered. The dynamics of the C 1 -interior of the set of diffeomorphisms possessing the shadowable measures is characterized as the uniform hyperbolicity. Full article
(This article belongs to the Special Issue Shadowing in Dynamical Systems)
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5 pages, 219 KiB  
Article
Equicontinuity, Expansivity, and Shadowing for Linear Operators
by Keonhee Lee and C. A. Morales
Axioms 2018, 7(4), 84; https://doi.org/10.3390/axioms7040084 - 15 Nov 2018
Cited by 1 | Viewed by 3230
Abstract
We prove that a linear operator of a complex Banach space has a shadowable point if and only if it has the shadowing property. In addition, every equicontinuous linear operator does not have the shadowing property and its spectrum is contained in the [...] Read more.
We prove that a linear operator of a complex Banach space has a shadowable point if and only if it has the shadowing property. In addition, every equicontinuous linear operator does not have the shadowing property and its spectrum is contained in the unit circle. Finally, we prove that if a linear operator is expansive and has the shadowing property, then the origin is the only nonwandering point. Full article
(This article belongs to the Special Issue Shadowing in Dynamical Systems)
7 pages, 245 KiB  
Article
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 3125
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)
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