Entropy of Singular Suspensions

Abstract

In this work, we investigate diffeomorphisms whose positiveness of topological entropy is destroyed by singular suspensions. We show that this phenomenon is rare in the set of $C^1$-diffeomorphisms. Precisely, we prove that for an open and dense set of $C^1$-diffeomorphism positive topological entropy is preserved by singular suspensions, even for suspensions with infinitely many singularities. We prove a similar result to the conservative diffeomorphisms. We apply our techniques to show that every expansive singular suspension $C^{1+\epsilon}$-flow over a three-dimensional manifold has positive topological entropy. Finally, we explore this phenomenon for Anosov dynamics, showing that to nullify the topological entropy for Anosov suspension flow, the set of singularities must capture the non-wandering set.