Transitivity and sensitivity of iterated function systems via Furstenberg families
- PDF / 381,733 Bytes
- 18 Pages / 439.37 x 666.142 pts Page_size
- 15 Downloads / 184 Views
Aequationes Mathematicae
Transitivity and sensitivity of iterated function systems via Furstenberg families Rahul Thakur and Ruchi Das
Abstract. In this paper, we study variants of transitivity and sensitivity via Furstenberg families for iterated function systems (IFSs). Using the concept of skew product transformation of an IFS, we obtain results related to characterizations of the concepts studied. Also results regarding the inverse, conjugacy and product of IFSs are studied. Wherever necessary examples and counterexamples are provided related to the results obtained. Mathematics Subject Classification. 37B05, 37B20, 37D45, 54H20. Keywords. Iterated function system, Transitivity, Mixing, Sensitivity, Product map.
1. Introduction In the theory of dynamical systems, chaos is an important phenomenon. Among the several definitions of chaos, Devaney’s definition which consists of three conditions namely transitivity, dense periodic points and sensitivity [8] is widely accepted. In the study of chaos, the concepts of transitivity and sensitivity are key ingredients. They are of great importance in the qualitative study of dynamical systems. The concept of transitivity can be traced back to Birkhoff [4]. It is also regarded as an irreducibility condition because one cannot decompose a transitive dynamical system into two disjoint sets with non-empty interiors so that they do not interact under the transformation. It is the basis for the study of several decomposition theorems. Sensitivity, also known as butterfly effect, characterizes the unpredictability of chaotic phenomena. If a dynamical system is sensitive, then a small change in the initial conditions leads to a significant change in the dynamics of the system. These days, the sensitivity theory of dynamical systems is being widely studied by many researchers [12,23]. While studying the transitivity or sensitivity of dynamical systems one may encounter a natural question: “Do they happen
R. Thakur, R. Das
AEM
at regular intervals of time?” Since a Furstenberg familiy is nothing but a collection of subsets of non-negative integers, one can also answer this question using Furstenberg families and classify transitive as well as sensitive dynamical systems. The study of transitivity via Furstenberg families has helped obtain some very useful and applicable results for dynamical systems [5,15]. In [26], the authors introduced and studied the concept of sensitivity via Furstenberg families. Since then many researchers have been working in this direction [24,27,28]. In [21], Shao studied the concepts of proximity, distality and recurrence via Furstenberg families. In [28], the authors studied the sensitivity on products of dynamical systems via Furstenberg families and obtained some sufficient conditions to ensure the sensitivity. Using the theory of Furstenberg families, several other chaotic properties have also been studied for autonomous as well as non-autonomous dynamical systems [16,18,22,25,29]. Motivated by the above works, in this paper we study the transitivity
Data Loading...
