Chaos for induced hyperspace maps

Abstract

Abstract. In this paper we wish to relate the dynamics of the base map to the dynamics of the induced map. In the process, we obtain conditions on the endowed hyperspace topology under which the chaotic behaviour of the map on the base space is inherited by the induced map on the hyperspace. Several of the known results come up as corollaries to our results. We also discuss some metric related dynamical properties on the hyperspace that cannot be deduced for the base dynamics.2000 AMS Classification: 54B20, 54C05

Induced dynamics on the hyperspaces

“…Thus, it is important to study the set-valued dynamics induced by a continuous self map which inturn can help characterizing the dynamics of a general dynamical system. Many researchers have addressed the problem and many of the questions in this direction have been answered [1,8,11,13,14,16]. In the process, the dynamical behavior of a system and its corresponding set-valued counterpart has been investigated and several interesting results have been obtained.…”

“…A special case when the family F is a singleton has been investigated by several authors and a lot of work in this direction has already been done [1,8,11,13,14,16]. It was proved that while weakly mixing and topological mixing on the two spaces are equivalent, transitivity on the base space need not imply transitivity on the hyperspace [1,16]. While [8] investigated the topological entropy of induced system of all nonempty compact subsets of X, [13] investigated the problem for a general hyperspace endowed with a general hyperspace topology.…”

“…In the process, the dynamical behavior of a system and its corresponding set-valued counterpart has been investigated and several interesting results have been obtained. In [1,16], authors proved that while weakly mixing and topological mixing on the two spaces are equivalent, transitivity on the base space need not imply transitivity on the hyperspace. Interesting results relating the topological entropy of the two spaces have been obtained [8].…”

F-sensitivity and (F_1, F_2)-sensitivity between dynamical systems and their induced hyperspace dynamical systems

“…It is well-known that (E, f) is topologically conjugated to the subsystem of (2 E , 2 f ) that consists of the singleton sets of E when E satisfies certain conditions and an appropriate hyperspace topology is selected, e.g., hit-or-miss topology (see [28][29][30]) or Vietoris topology (see [3,16,36]). Clearly, an invariant subset of the original system (E, f) becomes a fixed point of the hyperspace system (2 E , 2 f ).…”

On the dynamics of subcontinua of a tree

“…The natural question arises here: what is the connection between dynamical properties of the base map f and its extension F . For papers related to this topic, see [1], [5], [8], [11].…”