Topological dynamics on hyperspaces

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.

Keywords

• Hyperspace
• Hit and miss topology
• Hit and far-miss topology
• Induced map
• Transitivity
• Mixing
• Horseshoe
• Equicontinuity
• Scrambled set