Inventiones mathematicae

, Volume 128, Issue 2, pp 303-327

First online:

Harmonic measure on the Julia set for polynomial-like maps

  • Anna ZdunikAffiliated withInstitute of Mathematics, Warsaw University, ul. Banacha 2, 02-097 Warszawa, Poland

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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.