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

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 ${\mathrm{d}im}_H{\mu }_F>\log d\mathrm{/}{\lambda }_1+\log d\mathrm{/}{\lambda }_2$ when ${\lambda }_2>\log d$, and that ${\mathrm{d}im}_H{\mu }_F=2\log d\mathrm{/}{\lambda }_2$ when ${\lambda }_2\ge 2\log d$. 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.