  Given a polynomial $f\in\mathbb{C}[x]$, we consider the family of
superelliptic curves $y^d=f(x)$ and their Jacobians $J_d$ for varying integers
$d$. We show that for any integer $g$ the number of abelian varieties up to
isogeny of dimension $\le g$ which appear in any $J_d$ is finite and their
multiplicities are bounded.
