Dimension of equilibrium measures for complex maps

Abstract

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 $μ$ with Lyapunov exponents $χ_1\geq \ldots\geq χ_k$, we show $\dim_\mathrm{H}(μ) = \log d\sum_{i\leq k}\frac{1}{χ_i}$ where $\dim_\mathrm{H}(μ)$ is the Hausdorff dimension of the measure $μ$. This gives an answer to the question of Fornæss and Sibony, and proves the Binder-DeMarco Conjecture.