On the number of maximal chain transitive sets in fiber bundles

Abstract

This article studies chain transitivity for semigroup actions on fiber bundles whose typical fibers are compact topological spaces. We discuss the number of maximal chain transitive sets and, as a consequence, we obtain conditions for the existence of a finest Morse decomposition. Some of the results obtained are applied to orthonormal frame and Stiefel manifolds. A description of the maximal chain transitive sets is provided in terms of the action of shadowing semigroups. This description is well known in the literature under the hypothesis of local transitivity. Here, we exclude the hypothesis of local transitivity when the state space is a compact quotient space of a topological group.