Hypergraphs accumulate infinitely often

We show that the set ${\mathrm{\Pi }}^{(k)}$ of Turán densities of $k$-uniform hypergraphs has infinitely many accumulation points in $[0,1)$ for every $k\ge 3$. This extends an earlier result of ours showing that ${\mathrm{\Pi }}^{(k)}$ has at least one such accumulation point.