Inventiones mathematicae

, Volume 128, Issue 2, pp 303–327

Harmonic measure on the Julia set for polynomial-like maps


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

DOI: 10.1007/s002220050142

Cite this article as:
Zdunik, A. Invent math (1997) 128: 303. doi:10.1007/s002220050142


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.

Copyright information

© Springer-Verlag Berlin Heidelberg 1997