For certain families of complex maps, we give a formula for the Hausdorff dimension of the equilibrium measure. In particular, given an endomorphism $f$ of $\mathbb C\mathbb P^k$ of algebraic degree $d \ge2$, and given the equilibrium measure $\mu$ with Lyapunov exponents $\chi_1\geq \ldots\geq \chi_k$, we show $\dim_\mathrm{ H}(\mu) = \log d\sum_{ i\leq k}\frac{ 1}{ \chi_i}$ where $\dim_\mathrm{ H}(\mu)$ is the Hausdorff dimension of the measure $\mu$. This gives an answer to the question of Forn{ \ae}ss and Sibony, and proves the Binder-DeMarco Conjecture.
Categories: math.DS math.CV