scholarly journals Shadowing as a structural property of the space of dynamical systems

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jonathan Meddaugh
Keyword(s):  
Dynamical Systems ◽  
Large Class ◽  
Structural Property ◽  
Metric Spaces ◽  
Shadowing Property ◽  
Compact Metric Spaces

<p style='text-indent:20px;'>We demonstrate that there is a large class of compact metric spaces for which the shadowing property can be characterized as a structural property of the space of dynamical systems. We also demonstrate that, for this class of spaces, in order to determine whether a system has shadowing, it is sufficient to check that <i>continuously generated</i> pseudo-orbits can be shadowed.</p>

scholarly journals Ahlfors regularity and fractal dimension of Smale spaces

2021 ◽  
pp. 1-52
Author(s):  
DIMITRIS MICHAIL GERONTOGIANNIS
Keyword(s):  
Fractal Dimension ◽  
Markov Chains ◽  
Large Class ◽  
Metric Spaces ◽  
Topological Conjugacy ◽  
Compact Metric Spaces ◽  
Box Counting ◽  
Ahlfors Regularity ◽  
Smale Space ◽  
Smale Spaces

Abstract We prove that, up to topological conjugacy, every Smale space admits an Ahlfors regular Bowen measure. Bowen’s construction of Markov partitions implies that Smale spaces are factors of topological Markov chains. The latter are equipped with Parry’s measure, which is Ahlfors regular. By extending Bowen’s construction, we create a tool for transferring the Ahlfors regularity of the Parry measure down to the Bowen measure of the Smale space. An essential part of our method uses a refined notion of approximation graphs over compact metric spaces. Moreover, we obtain new estimates for the Hausdorff, box-counting and Assouad dimensions of a large class of Smale spaces.

scholarly journals Constructing minimal homeomorphisms on point-like spaces and a dynamical presentation of the Jiang–Su algebra

2018 ◽  
Vol 2018 (742) ◽  
pp. 241-261 ◽  
Author(s):  
Robin J. Deeley ◽  
Ian F. Putnam ◽  
Karen R. Strung
Keyword(s):  
Dynamical Systems ◽  
Inductive Limit ◽  
Metric Spaces ◽  
Lefschetz Number ◽  
Main Ingredient ◽  
Positive Role ◽  
Compact Metric Spaces ◽  
Minimal Homeomorphisms ◽  
Hopf Theorem ◽  
Prime Dimension

Abstract The principal aim of the present paper is to give a dynamical presentation of the Jiang–Su algebra. Originally constructed as an inductive limit of prime dimension drop algebras, the Jiang–Su algebra has gone from being a poorly understood oddity to having a prominent positive role in George Elliott’s classification programme for separable, nuclear {\mathrm{C}^{*}} -algebras. Here, we exhibit an étale equivalence relation whose groupoid {\mathrm{C}^{*}} -algebra is isomorphic to the Jiang–Su algebra. The main ingredient is the construction of minimal homeomorphisms on infinite, compact metric spaces, each having the same cohomology as a point. This construction is also of interest in dynamical systems. Any self-map of an infinite, compact space with the same cohomology as a point has Lefschetz number one. Thus, if such a space were also to satisfy some regularity hypothesis (which our examples do not), then the Lefschetz–Hopf Theorem would imply that it does not admit a minimal homeomorphism.

Properties of Dynamical Systems on Dendrites and Graphs

2021 ◽  
Vol 31 (07) ◽  
pp. 2150100
Author(s):  
Zdeněk Kočan ◽  
Veronika Kurková ◽  
Michal Málek
Keyword(s):  
Dynamical Systems ◽  
Topological Entropy ◽  
Metric Spaces ◽  
Limit Set ◽  
Open Problems ◽  
Homoclinic Trajectory ◽  
Compact Metric Spaces ◽  
Minimal Sets ◽  
Continuous Maps ◽  
Omega Limit Set

Dynamical systems generated by continuous maps on compact metric spaces can have various properties, e.g. the existence of an arc horseshoe, the positivity of topological entropy, the existence of a homoclinic trajectory, the existence of an omega-limit set containing two minimal sets and other. In [Kočan et al., 2014] we consider six such properties and survey the relations among them for the cases of graph maps, dendrite maps and maps on compact metric spaces. In this paper, we consider fourteen such properties, provide new results and survey all the relations among the properties for the case of graph maps and all known relations for the case of dendrite maps. We formulate some open problems at the end of the paper.

scholarly journals Rokhlin dimension for actions of residually finite groups

2017 ◽  
Vol 39 (8) ◽  
pp. 2248-2304 ◽  
Author(s):  
GÁBOR SZABÓ ◽  
JIANCHAO WU ◽  
JOACHIM ZACHARIAS
Keyword(s):  
Large Class ◽  
Finite Groups ◽  
Metric Spaces ◽  
Nilpotent Groups ◽  
Asymptotic Dimension ◽  
Finitely Generated ◽  
Residually Finite ◽  
Compact Metric Spaces ◽  
Finite Dimensional ◽  
Residually Finite Groups

We introduce the concept of Rokhlin dimension for actions of residually finite groups on $\text{C}^{\ast }$-algebras, extending previous such notions for actions of finite groups and the integers by Hirshberg, Winter and the third author. We are able to extend most of their results to a much larger class of groups: those admitting box spaces of finite asymptotic dimension. This latter condition is a refinement of finite asymptotic dimension and has not previously been considered. In a detailed study we show that finitely generated, virtually nilpotent groups have box spaces with finite asymptotic dimension, providing a large class of examples. We show that actions with finite Rokhlin dimension by groups with finite-dimensional box spaces preserve the property of having finite nuclear dimension when passing to the crossed product $\text{C}^{\ast }$-algebra. We then establish a relation between Rokhlin dimension of residually finite groups acting on compact metric spaces and amenability dimension of the action in the sense of Guentner, Willett and Yu. We show that for free actions of infinite, finitely generated, nilpotent groups on finite-dimensional spaces, both these dimensional values are finite. In particular, the associated transformation group$\text{C}^{\ast }$-algebras have finite nuclear dimension. This extends an analogous result about $\mathbb{Z}^{m}$-actions by the first author to a significantly larger class of groups, showing that a large class of crossed products by actions of such groups fall under the remit of the Elliott classification programme. We also provide results concerning the genericity of finite Rokhlin dimension, and permanence properties with respect to the absorption of a strongly self-absorbing $\text{C}^{\ast }$-algebra.

On the Entropy Points and Shadowing in Uniform Spaces

2018 ◽  
Vol 28 (12) ◽  
pp. 1850155 ◽  
Author(s):  
Seyyed Alireza Ahmadi ◽  
Xinxing Wu ◽  
Zonghong Feng ◽  
Xin Ma ◽  
Tianxiu Lu
Keyword(s):  
Dynamical Systems ◽  
Metric Spaces ◽  
Uniform Spaces ◽  
Shadowing Property

We introduce and study the topological concepts of entropy points, expansivity and shadowing property for dynamical systems on noncompact nonmetrizable spaces, which generalize the relevant concepts for metric spaces. We also obtain various properties on uniform entropy on noncompact nonmetrizable spaces. The main result is a theorem which yields a relation between topological shadowing property and positive uniform entropy.

Shadowing, ergodic shadowing and uniform spaces

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5117-5124 ◽  
Author(s):  
Seyyed Ahmadi
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Metric Spaces ◽  
Uniform Spaces ◽  
Shadowing Property ◽  
Chain Transitivity ◽  
Metrizable Spaces

We introduce and study the topological concepts of ergodic shadowing, chain transitivity and topological ergodicity for dynamical systems on non-compact non-metrizable spaces. These notions generalize the relevant concepts for metric spaces. We prove that a dynamical system with topological ergodic shadowing property is topologically chain transitive, and that topological chain transitivity together with topological shadowing property implies topological ergodicity.

The topology of attractors

1996 ◽  
Vol 16 (6) ◽  
pp. 1311-1322 ◽  
Author(s):  
Judy A. Kennedy
Keyword(s):  
Large Class ◽  
Pairwise Disjoint ◽  
Metric Spaces ◽  
Nonempty Interior ◽  
Hilbert Cube ◽  
Residual Set ◽  
Compact Metric Spaces ◽  
Infinite Collection ◽  
Chain Recurrent Set ◽  
Recurrent Set

AbstractWe prove for a large class of compact metric spaces, including those manifolds of dimension at least two, Hilbert cube manifolds, and homogeneous Menger manifolds, that ‘most’ self-homeomorphisms (in the sense of residual set of homeomorphisms) have certain properties. Specifically, if F: X → X is one of these homeomorphisms, then F admits• a dense, open wandering set;• a nowhere dense chain recurrent set;• an infinite collection of attractors (and repellers), each of which has nonempty interior and cannot be reduced to a ‘smallest’ attractor (or ‘largest’ repeller); and an uncountable collection of pairwise disjoint quasi-attractors.We also discuss the topology of the boundaries of attractors.

Stability of Lyapunov exponents

1991 ◽  
Vol 11 (3) ◽  
pp. 469-484 ◽  
Author(s):  
F. Ledrappier ◽  
L.-S. Young
Keyword(s):  
Dynamical Systems ◽  
Lyapunov Exponents ◽  
Metric Spaces ◽  
Regularity Conditions ◽  
Random Perturbations ◽  
Compact Metric Spaces ◽  
Small Random Perturbations ◽  
Volume Preserving

AbstractWe consider small random perturbations of matrix cocycles over Lipschitz homeomorphisms of compact metric spaces. Lyapunov exponents are shown to be stable provided that our perturbations satisfy certain regularity conditions. These results are applicable to dynamical systems, particularly to volume-preserving diffeomorphisms.

scholarly journals Existence of the Solutions of Nonlinear Fractional Differential Equations Using the Fixed Point Technique in Extended b-Metric Spaces

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 158
Author(s):  
Liliana Guran ◽  
Monica-Felicia Bota
Keyword(s):  
Fixed Point ◽  
Metric Spaces ◽  
Boundary Value ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Existence Of A Solution ◽  
Shadowing Property ◽  
Limit Shadowing ◽  
Type Boundary ◽  
Cyclic Type

The purpose of this paper is to prove fixed point theorems for cyclic-type operators in extended b-metric spaces. The well-posedness of the fixed point problem and limit shadowing property are also discussed. Some examples are given in order to support our results, and the last part of the paper considers some applications of the main results. The first part of this section is devoted to the study of the existence of a solution to the boundary value problem. In the second part of this section, we study the existence of solutions to fractional boundary value problems with integral-type boundary conditions in the frame of some Caputo-type fractional operators.

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