Dimension of equilibrium measures for complex maps

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 $d\ge 2$, and given the equilibrium measure $\mu$ with Lyapunov exponents ${\chi }_1\ge \dots \ge {\chi }_k$, we show ${\dim }_{\mathrm{H}}(\mu )=\log d{\sum }_{i\le k}\frac{1}{{\chi }_i}$ where ${\dim }_{\mathrm{H}}(\mu )$ is the Hausdorff dimension of the measure $\mu$. This gives an answer to the question of Fornæss and Sibony, and proves the Binder-DeMarco Conjecture.