History-dependent unstable directions and the Binder--DeMarco conjecture

Abstract

We establish dimension bounds and an exact dimension formula for equilibrium measures of degree $d$ holomorphic endomorphisms of $\mathbb P^2=\mathbb P^2(\mathbb C)$ satisfying suitable expansion and domination conditions. Write $\lambda_1>\lambda_2>0$ for their Lyapunov exponents. If two inverse histories ending at the same point determine different fast directions, we prove that ${\mathrm dim}_H \mu_F>\log d/\lambda_1+\log d/\lambda_2$ when $\lambda_2>\log d$, and that ${\mathrm dim}_H \mu_F=2\log d/\lambda_2$ when $\lambda_2\ge2\log d$. Applying these results to an explicit quadratic family, we disprove the Binder--DeMarco conjecture on a non-empty open set of holomorphic endomorphisms. The proof adapts the projection growth strategy of Li--Pan--Tong--Xu to the holomorphic setting, combining strong leaf geometry and the Ledrappier--Young theory for endomorphisms with a multidimensional extension of Wu's restricted sum estimate.