Relationship between topological entropy and topological horseshoes

Characterize the relationship between topological entropy and the existence of topological horseshoes for area-preserving homeomorphisms or diffeomorphisms of annuli, including whether the absence of a topological horseshoe implies zero topological entropy.

Background

The paper uses results about rotation numbers for zero-entropy area-preserving maps of annuli and notes that positive topological entropy forces the existence of a topological horseshoe. It also observes that the converse implication is not generally available.

The unresolved relationship is relevant to the dynamical framework underlying the paper: a stronger equivalence between entropy and horseshoes could clarify when rotation-number methods apply to annular invariant sets.