The dimension of the maximal measure for a polynomial map (Q801183)

\textit{H. Brolin} [Ark. Mat. 6, 103-144 (1965; Zbl 0127.034)] gave a dynamical construction for the equilibrium distribution on the Julia set (the closure of the repelling periodic points) of a polynomial map of the complex numbers. In the present paper it is shown that, provided the Julia set is connected and hyperbolic, this measure has Lyapunov exponent equal to the logarithm of the degree and its Hausdorff dimension is one; that is the measure is concentrated on a set of dimension one. This contrasts with \textit{D. Ruelle}'s calculation [Ergodic Theory Dyn. Syst. 2, 99-107 (1982; Zbl 0506.58024)] that the Julia set of \(z\mapsto z^ q- p\) has Hausdorff dimension \(1+| p|^ 2/4 \log q+... \).