Universal bounds on the entropy of toroidal attractors

A toroidal set is a compactum K ⊆ R³ which has a neighbourhood basis of solid tori. We study the topological entropy of toroidal attractors K , bounding it from below in terms of purely topological properties of K . In particular, we show that for a toroidal set K , either any smooth attracting dynamics on K has an entropy at least log 2 , or (up to continuation) K admits smooth attracting dynamics which are stationary (hence with a zero entropy).