Weakly Almost Periodic Points and Some Chaotic Properties of Dynamical Systems

2015 ◽  
Vol 25 (09) ◽  
pp. 1550115 ◽  
Author(s):  
Jiandong Yin ◽  
Zuoling Zhou
Keyword(s):  
Dynamical Systems ◽  
Metric Space ◽  
Sufficient Conditions ◽  
Periodic Points ◽  
Full Measure ◽  
Almost Periodic ◽  
Countable Base ◽  
Compact Metric Space ◽  
Continuous Map ◽  
Weakly Almost Periodic

Let X be a compact metric space and f : X → X be a continuous map. In this paper, ergodic chaos and strongly ergodic chaos are introduced, and it is proven that f is strongly ergodically chaotic if f is transitive but not minimal and has a full measure center. In addition, some sufficient conditions for f to be Ruelle–Takens chaotic are presented. For instance, we prove that f is Ruelle–Takens chaotic if f is transitive and there exists a countable base [Formula: see text] of X such that for each i > 0, the meeting time set N(Ui, Ui) for Ui with respect to itself has lower density larger than [Formula: see text].

scholarly journals Immediate conditional hyperbolicity in dynamical systems

1983 ◽  
Vol 3 (4) ◽  
pp. 627-647
Author(s):  
Joseph Rosenblatt ◽  
Richard Swanson
Keyword(s):  
Vector Bundle ◽  
Dynamical Systems ◽  
Compact Manifold ◽  
Hyperbolic Systems ◽  
Sufficient Conditions ◽  
Periodic Points ◽  
Almost Periodic ◽  
Uniquely Ergodic

AbstractFor many diffeomorphisms of a compact manifold X, eventual conditional hyperbolicity implies immediate conditional hyperbolicity in some (possibly new) Finsler structures. That is, if A and B are vector bundle isomorphisms over the mapping ƒ of the base X, such that uniformly on X, then there exist new norms for A and B such that uniformly on X, whenever the mapping ƒ satisfies the condition that there exist infinitely many N ≥ 1 such that any ƒ-invariant. For example, this condition on ƒ holds if any one of the following conditions holds: (1) ƒ is periodic; (2) ƒ is periodic on its non-wandering set; (3) ƒ has a finite non-wandering set (for example, ƒ is a Morse-Smale diffeomorphism); (4) ƒ is an almost periodic mapping of a connected base X; (5) ƒ is a mapping of the circle with no periodic points; or (6) ƒ and all its powers are uniquely ergodic. We consider various types of eventually conditionally hyperbolic systems and describe sufficient conditions on ƒ to have immediate conditional hyperbolicity of these systems in some new Finsler structures. Thus, for a sizable class of dynamical systems, we settle, in the affirmative, a question raised by Hirsch, Pugh, and Shub.

scholarly journals The chain recurrent set, attractors, and explosions

1985 ◽  
Vol 5 (3) ◽  
pp. 321-327 ◽  
Author(s):  
Louis Block ◽  
John E. Franke
Keyword(s):  
Compact Space ◽  
Metric Space ◽  
Periodic Points ◽  
Limit Set ◽  
Compact Metric Space ◽  
Continuous Map ◽  
Special Case ◽  
Equivalent Conditions ◽  
Chain Recurrent Set ◽  
Recurrent Set

AbstractCharles Conley has shown that for a flow on a compact metric space, a point x is chain recurrent if and only if any attractor which contains the & ω-limit set of x also contains x. In this paper we show that the same statement holds for a continuous map of a compact metric space to itself, and additional equivalent conditions can be given. A stronger result is obtained if the space is locally connected.It follows, as a special case, that if a map of the circle to itself has no periodic points then every point is chain recurrent. Also, for any homeomorphism of the circle to itself, the chain recurrent set is either the set of periodic points or the entire circle. Finally, we use the equivalent conditions mentioned above to show that for any continuous map f of a compact space to itself, if the non-wandering set equals the chain recurrent set then f does not permit Ω-explosions. The converse holds on manifolds.

On multi-sensitivity with respect to a vector

2018 ◽  
Vol 32 (15) ◽  
pp. 1850166 ◽  
Author(s):  
Lixin Jiao ◽  
Lidong Wang ◽  
Fengquan Li ◽  
Heng Liu
Keyword(s):  
Dynamical System ◽  
Metric Space ◽  
Hausdorff Metric ◽  
Sufficient Conditions ◽  
Compact Metric Space ◽  
Basic Properties ◽  
Continuous Map ◽  
Periodic Sets ◽  
Vector Formula ◽  
Compact Subsets

Consider the surjective continuous map [Formula: see text]: [Formula: see text] defined on a compact metric space X. Let [Formula: see text] be the space of all non-empty compact subsets of X equipped with the Hausdorff metric and define [Formula: see text]: [Formula: see text] by [Formula: see text] for any [Formula: see text]. In this paper, we introduce several stronger versions of sensitivities, such as multi-sensitivity with respect to a vector, [Formula: see text]-sensitivity, strong multi-sensitivity. We obtain some basic properties of the concepts of these sensitivities and discuss the relationships with other sensitivities for continuous self-map on [0,[Formula: see text]1]. Some sufficient conditions for a dynamical system to be [Formula: see text]-sensitive are presented. Also, it is shown that the strong multi-sensitivity of f implies that [Formula: see text] is [Formula: see text]-sensitive. In turn, the [Formula: see text]-sensitivity of [Formula: see text] implies that [Formula: see text] is [Formula: see text]-sensitive. In particular, it is proved that if [Formula: see text] is a multi-transitive map with dense periodic sets, then f is [Formula: see text]-sensitive. Finally, we give a multi-sensitive example which is not [Formula: see text]-sensitive.

On Generic Properties of Nonautonomous Dynamical Systems

2018 ◽  
Vol 28 (08) ◽  
pp. 1850102 ◽  
Author(s):  
Francisco Balibrea ◽  
Jaroslav Smítal ◽  
Marta Štefánková
Keyword(s):  
Dynamical Systems ◽  
Topological Entropy ◽  
Compact Space ◽  
Metric Space ◽  
Generic Properties ◽  
Compact Metric Space ◽  
Continuous Map ◽  
Nonautonomous Dynamical Systems ◽  
Uniformly Convergent ◽  
Generic System

We consider nonautonomous dynamical systems consisting of sequences of continuous surjective maps of a compact metric space [Formula: see text]. Let [Formula: see text], [Formula: see text] and [Formula: see text] denote the space of systems [Formula: see text], which are uniformly convergent, or equicontinuous, or pointwise convergent (to a continuous map), respectively. We show that for [Formula: see text], the generic system in any of the spaces has infinite topological entropy, while, if [Formula: see text] is the Cantor set, the generic system in any of the spaces has zero topological entropy. This shows, among others, that the general results of the above type for arbitrary compact space [Formula: see text] are difficult if not impossible.

Convergence of successive approximations to a stochastic fixed point

1980 ◽  
Vol 17 (1) ◽  
pp. 297-299
Author(s):  
Arun P. Sanghvi
Keyword(s):  
Fixed Point ◽  
Recent Result ◽  
Metric Space ◽  
Sufficient Conditions ◽  
Successive Approximations ◽  
Compact Metric Space

This paper describes some sufficient conditions that ensure the convergence of successive random applications of a family of mappings {Γα : α ∈ A} on a compact metric space (X, d) to a stochastic fixed point. The results are similar in spirit to a recent result of Yahav (1975).

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>

Semidirect Product Compactifications

1983 ◽  
Vol 35 (1) ◽  
pp. 1-32
Author(s):  
F. Dangello ◽  
R. Lindahl
Keyword(s):  
Direct Product ◽  
Semidirect Product ◽  
Topological Semigroup ◽  
Sufficient Conditions ◽  
Almost Periodic Functions ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Topological Semigroups ◽  
Weakly Almost Periodic ◽  
Necessary And Sufficient

1. Introduction. K. Deleeuw and I. Glicksberg [4] proved that if S and T are commutative topological semigroups with identity, then the Bochner almost periodic compactification of S × T is the direct product of the Bochner almost periodic compactifications of S and T. In Section 3 we consider the semidirect product of two semi topological semigroups with identity and two unital C*-subalgebras and of W(S) and W(T) respectively, where W(S) is the weakly almost periodic functions on S. We obtain necessary and sufficient conditions and for a semidirect product compactification of to exist such that this compactification is a semi topological semigroup and such that this compactification is a topological semigroup. Moreover, we obtain the largest such compactifications.

scholarly journals Surjunctivity and topological rigidity of algebraic dynamical systems

2017 ◽  
Vol 39 (3) ◽  
pp. 604-619 ◽  
Author(s):  
SIDDHARTHA BHATTACHARYA ◽  
TULLIO CECCHERINI-SILBERSTEIN ◽  
MICHEL COORNAERT
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Sufficient Conditions ◽  
Countable Group ◽  
Continuous Group ◽  
Continuous Map ◽  
Algebraic Dynamical Systems ◽  
Metrizable Group ◽  
Group Automorphisms

Let$X$be a compact metrizable group and let$\unicode[STIX]{x1D6E4}$be a countable group acting on$X$by continuous group automorphisms. We give sufficient conditions under which the dynamical system$(X,\unicode[STIX]{x1D6E4})$is surjunctive, i.e. every injective continuous map$\unicode[STIX]{x1D70F}:X\rightarrow X$commuting with the action of$\unicode[STIX]{x1D6E4}$is surjective.

scholarly journals Li-Yorke Sensitivity of Set-Valued Discrete Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Heng Liu ◽  
Fengchun Lei ◽  
Lidong Wang
Keyword(s):  
Metric Space ◽  
Short Proof ◽  
Hausdorff Metric ◽  
Discrete Systems ◽  
Compact Metric Space ◽  
Continuous Map ◽  
Nonempty Compact ◽  
Compact Subsets

Consider the surjective, continuous mapf:X→Xand the continuous mapf¯of𝒦(X)induced byf, whereXis a compact metric space and𝒦(X)is the space of all nonempty compact subsets ofXendowed with the Hausdorff metric. In this paper, we give a short proof that iff¯is Li-Yoke sensitive, thenfis Li-Yorke sensitive. Furthermore, we give an example showing that Li-Yorke sensitivity offdoes not imply Li-Yorke sensitivity off¯.

On the polynomial entropy for morse gradient systems

2019 ◽  
Vol 69 (3) ◽  
pp. 611-624
Author(s):  
Jelena Katić ◽  
Milan Perić
Keyword(s):  
Metric Space ◽  
Singular Set ◽  
Compact Metric Space ◽  
Easy Method ◽  
Gradient Systems ◽  
Continuous Map ◽  
Negative Gradient

Abstract We adapt the construction from [HAUSEUX, L.—LE ROUX, F.: Polynomial entropy of Brouwer homeomorphisms, arXiv:1712.01502 (2017)] to obtain an easy method for computing the polynomial entropy for a continuous map of a compact metric space with finitely many non-wandering points. We compute the maximal cardinality of a singular set of Morse negative gradient systems and apply this method to compute the polynomial entropy for Morse gradient systems on surfaces.

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