Chain transitivity and variations of the shadowing property

2014 ◽  
Vol 35 (7) ◽  
pp. 2044-2052 ◽  
Author(s):  
WILLIAM R. BRIAN ◽  
JONATHAN MEDDAUGH ◽  
BRIAN E. RAINES
Keyword(s):  
Arbitrary Sequence ◽  
Ramsey Property ◽  
Shadowing Property ◽  
Chain Transitivity ◽  
A Chain

We show that, under the assumption of chain transitivity, the shadowing property is equivalent to the thick shadowing property. We also show that, if ${\mathcal{F}}$ is a family with the Ramsey property, then an arbitrary sequence of points in a chain transitive space can be ${\it\varepsilon}$-shadowed (for any ${\it\varepsilon}$) on a set in ${\mathcal{F}}$.

Shadowing, thick sets and the Ramsey property

2015 ◽  
Vol 36 (5) ◽  
pp. 1582-1595 ◽  
Author(s):  
PIOTR OPROCHA
Keyword(s):  
Arbitrary Sequence ◽  
Ramsey Property ◽  
Shadowing Property ◽  
Full Characterization ◽  
Equivalent Properties ◽  
Chain Recurrent Set ◽  
Recurrent Set

We provide a full characterization of relations between the shadowing property and the thick shadowing property. We prove that they are equivalent properties for non-wandering systems, the thick shadowing property is always a consequence of the shadowing property, and the thick shadowing property on the chain-recurrent set and the thick shadowing property are the same properties. We also provide a full characterization of the cases when for any family ${\mathcal{F}}$ with the Ramsey property an arbitrary sequence of points can be ${\it\varepsilon}$-traced over a set from ${\mathcal{F}}$.

Shadowing and average shadowing properties for iterated function systems

2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Alireza Zamani Bahabadi
Keyword(s):  
Iterated Function System ◽  
Iterated Function Systems ◽  
Function System ◽  
Skew Product ◽  
Shadowing Property ◽  
Iterated Function ◽  
Chain Transitivity

AbstractIn this paper, we introduce the definitions of shadowing and average shadowing properties for iterated function systems and give some examples characterizing these definitions. We prove that an iterated function system has the shadowing property if and only if the step skew product corresponding to the iterated function system has the shadowing property. Also, we study some notions such as transitivity, chain transitivity, chain mixing and mixing for iterated function systems.

scholarly journals Variants of shadowing properties for iterated function systems on uniform spaces

Filomat ◽  
2021 ◽  
Vol 35 (8) ◽  
pp. 2565-2572
Author(s):  
Radhika Vasisht ◽  
Mohammad Salman ◽  
Ruchi Das
Keyword(s):  
Iterated Function System ◽  
Iterated Function Systems ◽  
Function System ◽  
Uniform Spaces ◽  
Shadowing Property ◽  
Topological Mixing ◽  
Topologically Transitive ◽  
Iterated Function ◽  
Chain Transitivity

In this paper, the notions of topological shadowing, topological ergodic shadowing, topological chain transitivity and topological chain mixing are introduced and studied for an iterated function system (IFS) on uniform spaces. It is proved that if an IFS has topological shadowing property and is topological chain mixing, then it has topological ergodic shadowing and it is topological mixing. Moreover, if an IFS has topological shadowing property and is topological chain transitive, then it is topologically ergodic and hence topologically transitive. Also, these notions are studied for the product IFS on uniform spaces.

scholarly journals G -Chain Mixing and G -Chain Transitivity in Metric G-Space

2022 ◽  
Vol 2022 ◽  
pp. 1-7
Author(s):  
Zhanjiang Ji
Keyword(s):  
Positive Integer ◽  
Shadowing Property ◽  
Open Set ◽  
Chain Transitivity ◽  
Dynamical Properties

Firstly, we introduce the concept of G -chain mixing, G -mixing, and G -chain transitivity in metric G -space. Secondly, we study their dynamical properties and obtain the following results. (1) If the map f has the G -shadowing property, then the map f is G -chain mixed if and only if the map f is G -mixed. (2) The map f is G -chain transitive if and only if for any positive integer k ≥ 2 , the map f k is G -chain transitive. (3) If the map f is G -pointwise chain recurrent, then the map f is G -chain transitive. (4) If there exists a nonempty open set U satisfying G U = U , U ¯ ≠ X , and f U ¯ ⊂ U , then we have that the map f is not G -chain transitive. These conclusions enrich the theory of G -chain mixing, G -mixing, and G -chain transitivity in metric G -space.

Some dynamical properties for free semigroup actions

2018 ◽  
Vol 18 (04) ◽  
pp. 1850032 ◽  
Author(s):  
Huihui Hui ◽  
Dongkui Ma
Keyword(s):  
Metric Space ◽  
Free Semigroup ◽  
Semigroup Action ◽  
Compact Metric Space ◽  
Shadowing Property ◽  
Semigroup Actions ◽  
Specification Property ◽  
Chain Transitivity ◽  
Weakly Mixing ◽  
Dynamical Properties

In this paper, we introduce the notions of weakly mixing and totally transitivity for a free semigroup action. Let [Formula: see text] be a free semigroup acting on a compact metric space generated by continuous open self-maps. Assuming shadowing for [Formula: see text] we relate the average shadowing property of [Formula: see text] to totally transitivity and its variants. Also, we study some properties such as mixing, shadowing and average shadowing properties, transitivity, chain transitivity, chain mixing and specification property for a free semigroup action.

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.

Observation of Atom Rows in Thorium Pyromellitate Using a Field Emission STEM

1977 ◽  
Vol 35 ◽  
pp. 80-81
Author(s):  
H. Todokoro ◽  
S. Nomura ◽  
T. Komoda
Keyword(s):  
Field Emission ◽  
Amorphous Carbon ◽  
Carbon Film ◽  
Dark Field Image ◽  
Dark Field ◽  
Amorphous Carbon Film ◽  
Atomic Size ◽  
Wide Range ◽  
A Chain

It is interesting to observe polymers at atomic size resolution. Some works have been reported for thorium pyromellitate by using a STEM (1), or a CTEM (2,3). The results showed that this polymer forms a chain in which thorium atoms are arranged. However, the distance between adjacent thorium atoms varies over a wide range (0.4-1.3nm) according to the different authors.The present authors have also observed thorium pyromellitate specimens by means of a field emission STEM, described in reference 4. The specimen was prepared by placing a drop of thorium pyromellitate in 10-3 CH3OH solution onto an amorphous carbon film about 2nm thick. The dark field image is shown in Fig. 1A. Thorium atoms are clearly observed as regular atom rows having a spacing of 0.85nm. This lattice gradually deteriorated by successive observations. The image changed to granular structures, as shown in Fig. 1B, which was taken after four scanning frames.

Time-Resolved Cryo-Electron Microscopy of Oscillating Microtubules

1990 ◽  
Vol 48 (1) ◽  
pp. 502-503
Author(s):  
Eva-Maria Mandelkow ◽  
Ron Milligan
Keyword(s):  
Electron Microscopy ◽  
Dynamic Instability ◽  
Entire Solution ◽  
Reaction Scheme ◽  
Time Resolved ◽  
X Ray ◽  
X Ray Scattering ◽  
A Chain ◽  
Ray Scattering

Microtubules form part of the cytoskeleton of eukaryotic cells. They are hollow libers of about 25 nm diameter made up of 13 protofilaments, each of which consists of a chain of heterodimers of α-and β-tubulin. Microtubules can be assembled in vitro at 37°C in the presence of GTP which is hydrolyzed during the reaction, and they are disassembled at 4°C. In contrast to most other polymers microtubules show the behavior of “dynamic instability”, i.e. they can switch between phases of growth and phases of shrinkage, even at an overall steady state [1]. In certain conditions an entire solution can be synchronized, leading to autonomous oscillations in the degree of assembly which can be observed by X-ray scattering (Fig. 1), light scattering, or electron microscopy [2-5]. In addition such solutions are capable of generating spontaneous spatial patterns [6].In an earlier study we have analyzed the structure of microtubules and their cold-induced disassembly by cryo-EM [7]. One result was that disassembly takes place by loss of protofilament fragments (tubulin oligomers) which fray apart at the microtubule ends. We also looked at microtubule oscillations by time-resolved X-ray scattering and proposed a reaction scheme [4] which involves a cyclic interconversion of tubulin, microtubules, and oligomers (Fig. 2). The present study was undertaken to answer two questions: (a) What is the nature of the oscillations as seen by time-resolved cryo-EM? (b) Do microtubules disassemble by fraying protofilament fragments during oscillations at 37°C?

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