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 C Pᵏ of algebraic degree d ≥2, and given the equilibrium measure μ with Lyapunov exponents χ₁≥ …≥ χₖ, we show H(μ) = log d∑i≤ k1/χᵢ where H(μ) is the Hausdorff dimension of the measure μ. This gives an answer to the question of Forn{}ss and Sibony, and proves the Binder-DeMarco Conjecture.