Furstenberg family and chaos

A Furstenberg family F is a family,consisting of some subsets of the set of positive integers,which is hereditary upwards,i.e.A?B and A∈F imply B∈F.For a given system(i.e.,a pair of a complete metric space and a continuous self-map of the space)and for a Furstenberg family F,the definition of F-scrambled pairs of points in the space has been given,which brings the well-known scrambled pairs in Li-Yorke sense and the scrambled pairs in distribution sense to be F-scrambled pairs corresponding respectively to suitable Furstenberg family F.In the present paper we explore the basic properties of the set of F-scrambled pairs of a system.The generically F-chaotic system and the generically strongly F-chaotic system are defined.A criterion for a generically strongly F-chaotic system is showed.

Chaotic Properties in the Sense of Furstenberg Families in Set-Valued Discrete Dynamical Systems

Let be a non-autonomous discrete system and be a set-valued discrete system induced by it. Where, is the space formed by all non-empty compact subsets of X endowed with the Hausdorff metric H, is a set-valued mapping sequence induced by . It is proved that is -chaos, then is -chaos. Where -chaos is denoted to -sensitive, -sensitive, -transitive, -accessible, -weakly mixing, -m-sensitive, infinitely sensitive, or syndetically transitive.

Relative Broken Family Sensitivity

Letπ:(X,T)→(Y,S)be a factor map between two topological dynamical systems,and F_(a) Furstenberg family of Z.We introduce the notion of relative broken F-sensitivity.Let Fs(resp.Fpubd,Finf)be the families consisting of all syndetic subsets(resp.positive upper Banach density subsets,infinite subsets).We show that for a factor mapπ:(X,T)→(Y,S)between transitive systems,πis relatively broken F-sensitive for F=Fs or Fpubd if and only if there exists a relative sensitive pair which is an F-recurrent point of(R_(π),T^((2)));is relatively broken Finf-sensitive if and only if there exists a relative sensitive pair which is not asymptotic.For a factor mapπ:(X,T)→(Y,S)between minimal systems,we get the structure of relative broken F-sensitivity by the factor map to its maximal equicontinuous factor.

ON F-SENSITIVE PAIRS

In the present paper, we define sensitive pairs via Furstenberg families and discuss the relation of three definitions： sensitivity, F -sensitivity and F -sensitive pairs, see Theorem 1. For transitive systems, we give some sufficient conditions to ensure the existence of F -sensitive pairs. In particular, each non-minimal E system （M system, P system） has positive lower density （ Fs , Fr resp.）-sensitive pairs almost everywhere. Moreover, each non-minimal M system is Fts -sensitive. Finally, by some examples we show that： （1） F -sensitivity can not imply the existence of F -sensitive pairs. That means there exists an F -sensitive system, which has no F -sensitive pairs. （2） There is no immediate relation between the existence of sensitive pairs and Li-Yorke chaos, i.e., there exists a system （X, f ） without Li-Yorke scrambled pairs, which has κ B -sensitive pairs almost everywhere. （3） If the system （G, f ） is sensitive, where G is a finite graph, then it has κ B -sensitive pairs almost everywhere.

A class of Furstenberg families and their applications to chaotic dynamics

For each sequence of positive real numbers,tending to positive infinity,a Furstenberg family is defined.All these Furstenberg families are compatible with dynamical systems.Then,chaos with respect to such Furstenberg families are intently discussed.This greatly improves some classica results of distributional chaos.To confirm the effectiveness of these improvements,the relevant examples are provided finally.

Recent Development of Chaos Theory in Topological Dynamics

We give a summary on the recent development of chaos theory in topological dynamics, focusing on Li-Yorke chaos, Devaney chaos, distributional chaos, positive topological entropy, weakly mixing sets and so on, and their relationships.

On multi-transitivity with respect to a vector

A topological dynamical system(X,f)is said to be multi-transitive if for every n∈N the system(Xn,f×f2××fn)is transitive.We introduce the concept of multi-transitivity with respect to a vector and show that multi-transitivity can be characterized by the hitting time sets of open sets,answering a question proposed by Kwietniak and Oprocha(2012).We also show that multi-transitive systems are Li-Yorke chaotic.

On λ-Power Distributional n-Chaos

For each real number λ∈ [0, 1], λ-power distributional chaos has been in- troduced and studied via Furstenberg families recently. The chaoticity gets stronger and stronger as A varies from 1 to 0, where 1-power distributional chaos is exactly the usual distributional chaos. As a generalization of distributional n-chaos,λ-power distributional n-chaos is defined similarly. Lots of classic results on distributional chaos can be improved to be the versions of λ-power distributional n-chaos accordingly. A practical method for distinguishing 0-power distributional n-chaos is given. A transitive system is constructed to be 0-power distributionally n-chaotic but without any distributionally （n ＋ 1）-scrambled tuples. For each λ∈ [0, 1], ）,-power distributional n-chaos can still appear in minimal systems with zero topological entropy.