scholarly journals The Research of Sequence Shadowing Property and Regularly Recurrent Point on the Double Inverse Limit Space

2022 ◽  
Vol 2022 ◽  
pp. 1-6
Author(s):  
Zhan jiang Ji
Keyword(s):  
Inverse Limit ◽  
Point Sets ◽  
Recurrent Point ◽  
Shadowing Property ◽  
Limit Space ◽  
Shift Map ◽  
Dynamical Properties ◽  
Definition Of ◽  
Inverse Limit Space

According to the definition of sequence shadowing property and regularly recurrent point in the inverse limit space, we introduce the concept of sequence shadowing property and regularly recurrent point in the double inverse limit space and study their dynamical properties. The following results are obtained: (1) Regularly recurrent point sets of the double shift map σ f ∘ σ g are equal to the double inverse limit space of the double self-map f ∘ g in the regularly recurrent point sets. (2) The double self-map f ∘ g has sequence shadowing property if and only if the double shift map σ f ∘ σ g has sequence shadowing property. Thus, the conclusions of sequence shadowing property and regularly recurrent point are generalized to the double inverse limit space.

scholarly journals The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action

2021 ◽  
Vol 19 (1) ◽  
pp. 1290-1298
Author(s):  
Zhanjiang Ji
Keyword(s):  
Group Action ◽  
Inverse Limit ◽  
The Self ◽  
Recurrent Point ◽  
Shadowing Property ◽  
Inverse Limit Spaces ◽  
Limit Space ◽  
Dynamical Properties ◽  
Inverse Limit Space

Abstract First, we give the concepts of G-sequence shadowing property, G-equicontinuity and G-regularly recurrent point. Second, we study their dynamical properties in the inverse limit space under group action. The following results are obtained. (1) The self-mapping f f has the G-sequence shadowing property if and only if the shift mapping σ \sigma has the G ¯ \overline{G} -sequence shadowing property; (2) The self-mapping f f is G-equicontinuous if and only if the shift mapping σ \sigma is G ¯ \overline{G} -equicontinuous; (3) R R G ¯ ( σ ) = lim ← ( R R G ( f ) , f ) R{R}_{\overline{G}}\left(\sigma )=\underleftarrow{\mathrm{lim}}\left(R{R}_{G}(f),f) . These conclusions make up for the lack of theory in the inverse limit space under group action.

scholarly journals Inverse Limit Spaces with Various Shadowing Property

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Ali Barzanouni
Keyword(s):  
Inverse Limit ◽  
Shadowing Property ◽  
Inverse Limit Spaces ◽  
Limit Space ◽  
Inverse Shadowing ◽  
Shift Map ◽  
Inverse Limit Space ◽  
The Relationship

We discuss the relationship between ergodic shadowing property and inverse shadowing property offand that of the shift map σfon the inverse limit space.

On shadowing property for inverse limit spaces

2008 ◽  
Vol 58 (1) ◽  
Author(s):  
Ekta Shah ◽  
T. Das
Keyword(s):  
Inverse Limit ◽  
Shadowing Property ◽  
Inverse Limit Spaces ◽  
Limit Space ◽  
Shift Map ◽  
Inverse Limit Space

AbstractWe study here the G-shadowing property of the shift map σ on the inverse limit space X f, generated by an equivariant self-map f on a metric G-space X.

Dynamical properties of the shift maps on the inverse limit spaces

1992 ◽  
Vol 12 (1) ◽  
pp. 95-108 ◽  
Author(s):  
Shihai Li
Keyword(s):  
Inverse Limit ◽  
Initial Conditions ◽  
Inverse Limit Spaces ◽  
Limit Space ◽  
Shift Map ◽  
Definition Of ◽  
Chainable Continua ◽  
Inverse Limit Space ◽  
Chain Recurrent Set ◽  
Recurrent Set

AbstractIn this paper, we prove the following results about the shift map on the inverse limit space of a compact metric space and a sole bonding map: (1) the chain recurrent set of the shift map equals the inverse limit space of the chain recurrent set of the sole bonding map. Similar results are proved for the nonwandering set, ω-limit set, recurrent set, and almost periodic set. (2) The shift map on the inverse limit space is chaotic in the sense of Devaney if and only if its sole bonding map is chaotic. (3) If the sole bonding map is ω-chaotic, then the shift map on the inverse limit space is ω-chaotic. With a modification of the definition of ω-chaos we show the converse is true. (4) We prove that a transitive map on a closed invariant set containing an interval must have sensitive dependence on initial conditions on the whole set. Then it is both ω-chaotic and chaotic in the sense of Devaney with chaotic set the whole set. At last we give a new method of constructing chaotic homeomorphisms on chainable continua, especially on pseudo-arcs.

scholarly journals G-Expansibility and G-Almost Periodic Point under Topological Group Action

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Zhanjiang Ji
Keyword(s):  
Topological Group ◽  
Periodic Point ◽  
Inverse Limit ◽  
The Self ◽  
Almost Periodic ◽  
Shadowing Property ◽  
Limit Shadowing ◽  
Shift Map ◽  
Inverse Limit Space ◽  
Almost Periodic Point

Firstly, the new concepts of G − expansibility, G − almost periodic point, and G − limit shadowing property were introduced according to the concepts of expansibility, almost periodic point, and limit shadowing property in this paper. Secondly, we studied their dynamical relationship between the self-map f and the shift map σ in the inverse limit space under topological group action. The following new results are obtained. Let X , d be a metric G − space and X f , G ¯ ,   d ¯ , σ be the inverse limit space of X , G , d , f . (1) If the map f : X ⟶ X is an equivalent map, then we have A P G ¯ σ = Lim ← A p G f , f . (2) If the map f : X ⟶ X is an equivalent surjection, then the self-map f is G − expansive if and only if the shift map σ is G ¯ − expansive. (3) If the map f : X ⟶ X is an equivalent surjection, then the self-map f has G − limit shadowing property if and only if the shift map σ has G ¯ − limit shadowing property. The conclusions of this paper generalize the corresponding results given in the study by Li, Niu, and Liang and Li . Most importantly, it provided the theoretical basis and scientific foundation for the application of tracking property in computational mathematics and biological mathematics.

Dynamical properties of shift maps on inverse limits with a set valued function

2016 ◽  
Vol 38 (4) ◽  
pp. 1499-1524 ◽  
Author(s):  
JUDY KENNEDY ◽  
VAN NALL
Keyword(s):  
Topological Structure ◽  
Inverse Limit ◽  
Natural Generalization ◽  
Continuous Functions ◽  
Invariant Sets ◽  
Inverse Limits ◽  
Limit Space ◽  
Shift Map ◽  
Single Well ◽  
Inverse Limit Space

Set-valued functions from an interval into the closed subsets of an interval arise in various areas of science and mathematical modeling. Research has shown that the dynamics of a single-valued function on a compact space are closely linked to the dynamics of the shift map on the inverse limit with the function as the sole bonding map. For example, it has been shown that with Devaney’s definition of chaos the bonding function is chaotic if and only if the shift map is chaotic. One reason for caring about this connection is that the shift map is a homeomorphism on the inverse limit, and therefore the topological structure of the inverse-limit space must reflect in its richness the dynamics of the shift map. In the set-valued case there may not be a natural definition for chaos since there is not a single well-defined orbit for each point. However, the shift map is a continuous single-valued function so it together with the inverse-limit space form a dynamical system which can be chaotic in any of the usual senses. For the set-valued case we demonstrate with theorems and examples rich topological structure in the inverse limit when the shift map is chaotic (on certain invariant sets). We then connect that chaos to a property of the set-valued function that is a natural generalization of an important chaos producing property of continuous functions.

scholarly journals Sensitivity for topologically double ergodic dynamical systems

2021 ◽  
Vol 6 (10) ◽  
pp. 10495-10505
Author(s):  
Risong Li ◽  
◽  
Xiaofang Yang ◽  
Yongxi Jiang ◽  
Tianxiu Lu ◽  
...  
Keyword(s):  
Dynamical Systems ◽  
Metric Space ◽  
Inverse Limit ◽  
Compact Metric Space ◽  
Limit Space ◽  
Factor Map ◽  
Continuous Surjection ◽  
Inverse Limit Space

<abstract><p>As a stronger form of multi-sensitivity, the notion of ergodic multi-sensitivity (resp. strongly ergodically multi-sensitivity) is introduced. In particularly, it is proved that every topologically double ergodic continuous selfmap (resp. topologically double strongly ergodic selfmap) on a compact metric space is ergodically multi-sensitive (resp. strongly ergodically multi-sensitive). And for any given integer $ m\geq 2 $, $ f $ is ergodically multi-sensitive (resp. strongly ergodically multi-sensitive) if and only if so is $ f^{m} $. Also, it is shown that if $ f $ is a continuous surjection, then $ f $ is ergodically multi-sensitive (resp. strongly ergodically multi-sensitive) if and only if so is $ \sigma_{f} $, where $ \sigma_{f} $ is the shift selfmap on the inverse limit space $ \lim\limits_{\leftarrow}(X, f) $. Moreover, it is proved that if $ f:X\rightarrow X $ (resp. $ g:Y\rightarrow Y $) is a map on a nontrivial metric space $ (X, d) $ (resp. $ (Y, d') $), and $ \pi $ is a semiopen factor map between $ (X, f) $ and $ (Y, g) $, then the ergodic multi-sensitivity (resp. the strongly ergodic multi-sensitivity) of $ g $ implies the same property of $ f $.</p></abstract>

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