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

TL;DR

This paper investigates equilibrium measures on saddle sets of holomorphic maps on P^2, deriving formulas for their dimensions and measures, especially for minimal saddle sets, linking dynamics, geometry, and invariant measures.

Contribution

It introduces new formulas for the Hausdorff and pointwise dimensions of equilibrium measures on saddle sets of holomorphic maps, including cases with constant preimage counting functions.

Findings

Derived estimates for pointwise dimension involving Lyapunov exponents.

Established formulas for Hausdorff dimension of equilibrium measures.

Proved that certain invariant measures are measures of maximal entropy on saddle sets.

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 $\mu_\phi$ associated to a class of H\"older potentials $\phi$ on $\Lambda$, and find the measures $\mu_\phi$ of iterates of arbitrary Bowen balls. Estimates for the pointwise dimension $\delta_{\mu_\phi}$ of $\mu_\phi$ that involve Lyapunov exponents and a correction term are found, and also a formula for the Hausdorff dimension of $\mu_\phi$ 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 $\nu$.