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 $d\ge 2$, and given the equilibrium measure $\hat{I}¼$ with Lyapunov exponents $\ddot{I}{‡}_1\ge \dots \ge \ddot{I}{‡}_k$, we show ${\dim }_{\mathrm{H}}(\hat{I}¼)=\log d{\sum }_{i\le k}\frac{1}{\ddot{I}{‡}_i}$ where ${\dim }_{\mathrm{H}}(\hat{I}¼)$ is the Hausdorff dimension of the measure $\hat{I}¼$. This gives an answer to the question of Fornæss and Sibony, and proves the Binder-DeMarco Conjecture.

We found a gap in the proof of one of the lemmas, on which we are working to fix