Inducing sensitivity on hyperspaces

Functional envelope of a non-autonomous discrete system

“…It can to see that f i , i = 0, 1 is not equicontinuous [7], but f 1 ∘ f 0 = id X and the system (X, F = {fn} ∞ n=0 ) is equicontinuous. Since S(X) is not compact in general, the orbit closure O G (φ) may be non-compact.…”

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.…”

“…They also investigated notions like sensitive dependence on initial conditions, topological entropy, Li-Yorke chaos, existence of Li-Yorke pairs, existence of horseshoe and the corresponding results were established. Investigating the inverse of the problem stated, they also investigated the behavior of an individual component of the system, given the dynamical behavior of the induced system on the hyperspace [13,14].…”

“…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].…”

“…They discussed various dynamical notions like dense periodicity, transitivity, weakly mixing, topological mixing and topological entropy. They also investigated metric related dynamical notions like equicontinuity, sensitivity, strong sensitivity and Li-Yorke chaoticity [13,14]. Authors extended their studies to the general case when the dynamics on X is generated by a finite family and derived the necessary and sufficient conditions for the induced system to exhibit various dynamical notions [11].…”