The Dimension of the Maximal Measure for a Polynomial Map

Let f: C -> C be a polynomial of degree d ? 2. The most interesting dynamics of f takes place on its Julia set J(f), an invariant compact perfect set which can be defined, for example, as the closure of the repelling periodic points. Brolin constructed the equilibrium distribution m on J(f), an ergodic measure which is also the weak limit as n x-> of a probability measure distributed equally over the dn nth inverse images of almost any point. We shall prove the following: