  Given a multimodal interval map $f:I \to I$ and a H\"older potential $\phi:I
\to \mathbb{R}$, we study the dimension spectrum for equilibrium states of
$\phi$. The main tool here is inducing schemes, used to overcome the presence
of critical points. The key issue is to show that enough points are `seen' by a
class of inducing schemes. We also compute the Lyapunov spectrum. We obtain the
strongest results when $f$ is a Collet-Eckmann map, but our analysis also holds
for maps satisfying much weaker growth conditions.
