Topological entropy and chaos for maps induced on hyperspaces

On the dynamics of subcontinua of a tree

“…In [12] it was proved that topological entropy of an interval dynamical system is equal to that of its connected envelope. In [11] the same was proved for transitive systems on graphs. Our next theorem establishes this equality for any dynamical system on a tree.…”

“…For interval maps such a characterization was known (see for instance [12]) and for transitive graph maps similar result was recently proved in [11]. Still for general graph maps the situation is unclear.…”

“…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].…”

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.…”

“…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]. In [13], Sharma and Nagar investigated some of the natural questions arising from this setting.…”