  A toroidal set is a compactum $K \subseteq \mathbb{R}^3$ 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).
