A compact set $K \subseteq \mathbb{R}^3$ is called \emph{toroidal} if it has a neighbourhood basis of solid tori. This is a natural generalization of the well-known notion of a cellular set. To any toroidal set one can assign a finite set of prime numbers called its prime divisors. These reflect purely topological properties of $K$.

Suppose $f$ is a diffeomorphism of $\mathbb{R}^3$ and $K$ is an attractor for $f$ which happens to be a toroidal set (solenoids are the canonical example of this). We prove the following: the entropy of $f$ on $K$ is bounded below by ${\rm log}(p_1 \cdot \ldots \cdot p_n)$, where the $p_i$ are the prime divisors of $K$. Since the latter depend only on $K$, this provides a universal lower bound on any $\mathcal{C}^{\infty}$ attracting dynamics on $K$.

In the talk we will discuss the geometric techniques used to prove this and discuss its (plausible?) validity when $f$ is just a homeomorphism.