Special Issue "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 8875

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
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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

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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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Published Papers (6 papers)

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Research

Article
Generic Homeomorphisms with Shadowing of One-Dimensional Continua
Axioms 2019, 8(2), 66; https://doi.org/10.3390/axioms8020066 - 26 May 2019
Viewed by 1099
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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Article
A Note on Anosov Homeomorphisms
Axioms 2019, 8(2), 54; https://doi.org/10.3390/axioms8020054 - 01 May 2019
Cited by 3 | Viewed by 1724
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)
Article
Relations between Shadowing and Inverse Shadowing in Dynamical Systems
Axioms 2019, 8(1), 11; https://doi.org/10.3390/axioms8010011 - 17 Jan 2019
Cited by 1 | Viewed by 1340
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)
Article
Diffeomorphisms with Shadowable Measures
Axioms 2018, 7(4), 93; https://doi.org/10.3390/axioms7040093 - 07 Dec 2018
Cited by 4 | Viewed by 1569
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 C1-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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Article
Equicontinuity, Expansivity, and Shadowing for Linear Operators
Axioms 2018, 7(4), 84; https://doi.org/10.3390/axioms7040084 - 15 Nov 2018
Cited by 1 | Viewed by 1516
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)
Article
A Type of the Shadowing Properties for Generic View Points
Axioms 2018, 7(1), 18; https://doi.org/10.3390/axioms7010018 - 20 Mar 2018
Cited by 3 | Viewed by 1400
Abstract
We show that if a C1 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 C1 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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