Multifractal analysis for multimodal maps

Given a multimodal interval map $f:I\to I$ and a H"older potential $\phi:I \to $\mathbb{R}$$,westudythedimensionspectrumforequilibriumstatesof$\phi$. 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.