Brownian motion and the equilibrium measure on the Julia set of a rational mapping (Q1203662)

The complex analytic dynamics for rational mappings \(Q(z) = P_ 1(z)/P_ 2(z)\) of degree \(d\geq 2\) is considered. The Julia set \(\mathcal J\) of \(Q\) is the set of points \(z\in\overline{\mathbb{C}}\) for which \(\{Q^ n(z)\}_{n \geq 1}\), \(Q^ 0(z) = z\), \(Q^{n+1}(z) =Q(Q^ n(z))\), is not a normal family in any neighborhood of \(z\). The Julia set is a nonempty, compact set satisfying \({\mathcal J} = Q({\mathcal J}) = Q^{-1}(\mathcal J)\). Certain ergodic properties of the equilibrium measure \(\nu\) of \(\mathcal J\) for mappings \(Q\) such that \(Q(\infty) = \infty\) and \(\infty \in \overline{\mathbb{C}}\setminus{\mathcal J}\) are investigated. It is proved that if a rational mapping \(Q\) has \(\infty\) as a fixed point in \(\overline{\mathbb{C}}\setminus{\mathcal J}\), then \(\mathcal J\) has positive capacity and the equilibrium measure is invariant. If \(\infty\) is attracting or superattracting, then the equilibrium measure is strongly mixing, whereas if \(\infty\) is neutral, then the equilibrium measure is ergodic and has entropy zero. The proofs use an induced action by the \(Q\) on the space of Brownian paths started at \(\infty\).