# Rigidity of harmonic measure

I. Popovici; Alexander Volberg

Fundamenta Mathematicae (1996)

- Volume: 150, Issue: 3, page 237-244
- ISSN: 0016-2736

## Access Full Article

top## Abstract

top## How to cite

topPopovici, I., and Volberg, Alexander. "Rigidity of harmonic measure." Fundamenta Mathematicae 150.3 (1996): 237-244. <http://eudml.org/doc/212174>.

@article{Popovici1996,

abstract = {Let J be the Julia set of a conformal dynamics f. Provided that f is polynomial-like we prove that the harmonic measure on J is mutually absolutely continuous with the measure of maximal entropy if and only if f is conformally equivalent to a polynomial. This is no longer true for generalized polynomial-like maps. But for such dynamics the coincidence of classes of these two measures turns out to be equivalent to the existence of a conformal change of variable which reduces the dynamical system to another one for which the harmonic measure equals the measure of maximal entropy.},

author = {Popovici, I., Volberg, Alexander},

journal = {Fundamenta Mathematicae},

keywords = {polynomial-like maps; measure of maximum entropy; harmonic measure},

language = {eng},

number = {3},

pages = {237-244},

title = {Rigidity of harmonic measure},

url = {http://eudml.org/doc/212174},

volume = {150},

year = {1996},

}

TY - JOUR

AU - Popovici, I.

AU - Volberg, Alexander

TI - Rigidity of harmonic measure

JO - Fundamenta Mathematicae

PY - 1996

VL - 150

IS - 3

SP - 237

EP - 244

AB - Let J be the Julia set of a conformal dynamics f. Provided that f is polynomial-like we prove that the harmonic measure on J is mutually absolutely continuous with the measure of maximal entropy if and only if f is conformally equivalent to a polynomial. This is no longer true for generalized polynomial-like maps. But for such dynamics the coincidence of classes of these two measures turns out to be equivalent to the existence of a conformal change of variable which reduces the dynamical system to another one for which the harmonic measure equals the measure of maximal entropy.

LA - eng

KW - polynomial-like maps; measure of maximum entropy; harmonic measure

UR - http://eudml.org/doc/212174

ER -

## References

top- [B] Z. Balogh, Rigidity of harmonic measure on totally disconnected fractals, preprint, Michigan State University, April 1995.
- [BPV] Z. Balogh, I. Popovici and A. Volberg, Conformally maximal polynomial-like dynamics and invariant harmonic measure, preprint, Uppsala Univ., U.U.D.M. Report 1994:32.
- [BV] Z. Balogh et A. Volberg, Principe de Harnack à la frontière pour des répulseurs holomorphes non récurrents, C. R. Acad. Sci. Paris 319 (1994), 351-354.
- [Br] H. Brolin, Invariant sets under iteration of rational functions, Ark. Mat. 6 (1965), 103-145. Zbl0127.03401
- [DH] A. Douady and F. Hubbard, On the dynamics of polynomial like mappings, Ann. Sci. Ecole Norm. Sup. 18 (1985), 287-345. Zbl0587.30028
- [Fo] S. R. Foguel, The Ergodic Theory of Markov Processes, Math. Stud. 21, Van Nostrand, New York, 1969.
- [FLM] A. Freire, A. Lopes and R. Ma né, An invariant measure for rational maps, Bol. Soc. Brasil Mat. 14 (1983), 45-62.
- [Lo] A. Lopes, Equilibrium measure for rational functions, Ergodic Theory Dynam. Systems 6 (1986), 393-399.
- [Ly1] M. Lyubich, Entropy of the analytic endomorphisms of the Riemann sphere, Funktsional. Anal. i Prilozhen. 15 (4) (1981), 83-84 (in Russian).
- [Ly2] M. Lyubich, Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory Dynam. Systems 3 (1983), 351-386.
- [LV] M. Yu. Lyubich and A. Volberg, A comparison of harmonic and maximal measures on Cantor repellers, J. Fourier Anal. Appl. 1 (1995), 359-379.
- [M] R. Ma né, On the uniqueness of the maximizing measure for rational maps, Bol. Soc. Brasil. Mat. 14 (1983), 27-43.
- [MR] R. Ma né and L. F. da Rocha, Julia sets are uniformly perfect, Proc. Amer. Math. Soc. 116 (1992), 251-257. Zbl0763.30010
- [SS] M. Shub and D. Sullivan, Expanding endomorphisms of the circle revisited, Ergodic Theory Dynam. Systems 5 (1985), 285-289. Zbl0583.58022
- [Z1] A. Zdunik, Parabolic orbifolds and the dimension of the maximal measure for rational maps, Invent. Mat. 99 (1990), 627-649. Zbl0820.58038
- [Z2] A. Zdunik, Invariant measure in the class of harmonic measures for polynomial-like mappings, preprint 538, Inst. Math., Polish Acad. Sci., 1995.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.