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>

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.

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 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.

Characterizations of topological dynamical systems whose transformation group C*-algebras are antiliminal and of type 1

1993 ◽  
Vol 13 (1) ◽  
pp. 1-5 ◽  
Author(s):  
Nobuo Aoki ◽  
Jun Tomiyama
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Metric Space ◽  
Transformation Group ◽  
Irreducible Representations ◽  
Topological Dynamical System ◽  
Compact Metric Space ◽  
C Algebras ◽  
Topological Dynamical Systems

AbstractFor a topological dynamical system Σ = (X, σ) where X is a compact metric space with a single homeomorphism σ, we determine the largest postliminal ideal of the transformation group C*-algebra A(Σ) as the intersection of all kernels of irreducible representations of A(Σ) induced from those recurrent points which are not periodic. The result implies characterizations of topological dynamical systems whose transformation group C*-algebras are anti-liminal and post-liminal, that is, of type 1.

scholarly journals ATTRACTOR–REPELLER PAIR, MORSE DECOMPOSITION AND LYAPUNOV FUNCTION FOR RANDOM DYNAMICAL SYSTEMS

2008 ◽  
Vol 08 (04) ◽  
pp. 625-641 ◽  
Author(s):  
ZHENXIN LIU ◽  
SHUGUAN JI ◽  
MENGLONG SU
Keyword(s):  
Dynamical Systems ◽  
Lyapunov Function ◽  
Metric Space ◽  
Stability Theory ◽  
Lyapunov Functions ◽  
Random Dynamical Systems ◽  
Morse Decomposition ◽  
Compact Metric Space ◽  
Morse Decompositions ◽  
The Stability

In the stability theory of dynamical systems, Lyapunov functions play a fundamental role. In this paper, we study the attractor–repeller pair decomposition and Morse decomposition for compact metric space in the random setting. In contrast to [7,17], by introducing slightly stronger definitions of random attractor and repeller, we characterize attractor–repeller pair decompositions and Morse decompositions for random dynamical systems through the existence of Lyapunov functions. These characterizations, we think, deserve to be known widely.

A COMPUTATIONAL ERGODIC THEOREM FOR INFINITE ITERATED FUNCTION SYSTEMS

2008 ◽  
Vol 08 (03) ◽  
pp. 365-381 ◽  
Author(s):  
NGUYEN DINH CONG ◽  
DOAN THAI SON ◽  
STEFAN SIEGMUND
Keyword(s):  
Dynamical Systems ◽  
Euclidean Space ◽  
Ergodic Theorem ◽  
Error Bound ◽  
Metric Space ◽  
Iterated Function Systems ◽  
Random Dynamical Systems ◽  
Compact Metric Space ◽  
Iterated Function ◽  
Time Average

Iterated function systems are examples of random dynamical systems and became popular as generators of fractals like the Sierpinski Gasket and the Barnsley Fern. In this paper we prove an ergodic theorem for iterated function systems which consist of countably many functions and which are contractive on average on an arbitrary compact metric space and we provide a computational version of this ergodic theorem in Euclidean space which allows to numerically approximate the time average together with an explicit error bound. The results are applied to an explicit example.

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