Almost Minimal Systems and Periodicity in Hyperspaces

TL;DR

This paper investigates the periodic points of induced maps on hyperspaces of compact metric spaces, providing conditions on admissible periods, constructing a specific minimal homeomorphism, and describing admissible periods for interval and symmetric product maps.

Contribution

It introduces new necessary conditions for admissible periods of hyperspace maps, constructs an almost totally minimal homeomorphism of the Cantor set, and characterizes admissible periods for interval and symmetric product induced maps.

Findings

Necessary conditions on admissible periods for hyperspace maps

Construction of an almost totally minimal homeomorphism of the Cantor set

Full description of admissible periods for interval and symmetric product maps

Abstract

Given a self-map of a compact metric space $X$, we study periodic points of the map induced on the hyperspace of closed subsets of $X$. We give some necessary conditions on admissible sets of periods for these maps. Seemingly unrelated to this, we construct an almost totally minimal homeomorphism of the Cantor set. We also apply our theory to give a full description of admissible period sets for induced maps of the interval maps. The description of admissible periods is also given for maps induced on symmetric products.