Harmonic measure on the Julia set for polynomial-like maps

Abstract.

For a generalized polynomial-like mapping we prove the existence of an invariant ergodic measure equivalent to the harmonic measure on the Julia set J( f).

We also prove that for polynomial-like mappings the harmonic measure is equivalent to the maximal entropy measure iff f is conformally equivalent to a polynomial.

Next, we show that the Hausdorff dimension of harmonic measure on the Julia set of a generalized polynomial-like map is strictly smaller than 1 unless the Julia set is connected.