Paper detail

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

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