Equilibrium measures on saddle sets of holomorphic maps on P^2

Abstract

We consider the case of hyperbolic basic sets $\Lambda$ of saddle type for holomorphic maps $f:{\mathbb{P}}^2\mathbb{C}\to {\mathbb{P}}^2\mathbb{C}$. We study equilibrium measures ${\mu }_{\varphi }$ associated to a class of Hölder potentials $\varphi$ on $\Lambda$, and find the measures ${\mu }_{\varphi }$ of iterates of arbitrary Bowen balls. Estimates for the pointwise dimension ${\delta }_{{\mu }_{\varphi }}$ of ${\mu }_{\varphi }$ that involve Lyapunov exponents and a correction term are found, and also a formula for the Hausdorff dimension of ${\mu }_{\varphi }$ in the case when the preimage counting function is constant on $\Lambda$. For terminal/minimal saddle sets we prove that an invariant measure $\nu$ obtained as a wedge product of two positive closed currents, is in fact the measure of maximal entropy for the \textit{restriction} $f{|}_{\Lambda }$. This allows then to obtain formulas for the measure $\nu$ of arbitrary balls, and to give a formula for the pointwise dimension and the Hausdorff dimension of $\nu$.