Sensitivity of iterated function systems

Abstract

The present work is concerned with the eqiucontinuity and sensitivity of iterated function systems (IFSs). Here, we consider more general case of IFSs, i.e. the IFSs generated by a family of relations. We generalize the concepts of transitivity, sensitivity and equicontinuity to these kinds of systems. This note investigates the relationships between these concepts. Then, several sufficient conditions for sensitivity of IFSs are presented. We introduce the notion of weak topologically exact for IFSs generated by a family of relations. It is proved that non-minimal weak topologically exact IFSs are sensitive. That yields to different examples of non-minimal sensitive systems which are not an M -system. Moreover, some interesting examples are given which provide some facts about the sensitive property of IFSs.

Distributional chaos in random dynamical systems

“…where I f and I g are as defined in (7). If ω ∈ A n 3 , then, from the definition of the set, there are two g's and one f among ω n−2 , ω n−1 , and ω n .…”

“…3,5,14], and results concerning chaos are not common. In [8], topological entropy was studied, and recently, some other chaotic notions in IFS were investigated in [1] and [7] (but randomness was not taken into account in these studies).…”

Finite Chaoticity and Pairwise Sensitivity of a Strong-Mixing Measure-Preserving Semi-Flow

“…One of the necessary conditions for various definitions of chaos [4][5][6][7][8] is sensitivity, which characterizes the unpredictability of chaos in dynamical systems, as is widely acknowledged. In recent years, there has been a widespread interest in the concept of sensitive dependence on initial conditions [5,[9][10][11][12][13], which has been formalized in various ways by several authors [14][15][16][17][18].…”

Collective Sensitivity, Collective Accessibility, and Collective Kato’s Chaos in Duopoly Games

“…In particular, they considered the F -transitivity, F -mixing, F -sensitivity, F -collective sensitivity, F -synchronous sensitivity, (F 1 ,F 2 )-sensitivity, and F -multi-sensitivity for a nonautonomous discrete dynamical system and extended the responding results of [27]. In [29], F. Ghane et al considered the equicontinuity and sensitivity of iterated function systems (for short, IFSs). In particular, they discussed more general case of IFSs (in other words, the IFSs which are generated by a family of relations).…”