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.

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.

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.

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.