Rigidity of harmonic measure

I. Popovici; Alexander Volberg

Fundamenta Mathematicae (1996)

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

Abstract

top
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.

How to cite

top

Popovici, 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
  1. [B] Z. Balogh, Rigidity of harmonic measure on totally disconnected fractals, preprint, Michigan State University, April 1995. 
  2. [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. 
  3. [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. 
  4. [Br] H. Brolin, Invariant sets under iteration of rational functions, Ark. Mat. 6 (1965), 103-145. Zbl0127.03401
  5. [DH] A. Douady and F. Hubbard, On the dynamics of polynomial like mappings, Ann. Sci. Ecole Norm. Sup. 18 (1985), 287-345. Zbl0587.30028
  6. [Fo] S. R. Foguel, The Ergodic Theory of Markov Processes, Math. Stud. 21, Van Nostrand, New York, 1969. 
  7. [FLM] A. Freire, A. Lopes and R. Ma né, An invariant measure for rational maps, Bol. Soc. Brasil Mat. 14 (1983), 45-62. 
  8. [Lo] A. Lopes, Equilibrium measure for rational functions, Ergodic Theory Dynam. Systems 6 (1986), 393-399. 
  9. [Ly1] M. Lyubich, Entropy of the analytic endomorphisms of the Riemann sphere, Funktsional. Anal. i Prilozhen. 15 (4) (1981), 83-84 (in Russian). 
  10. [Ly2] M. Lyubich, Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory Dynam. Systems 3 (1983), 351-386. 
  11. [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. 
  12. [M] R. Ma né, On the uniqueness of the maximizing measure for rational maps, Bol. Soc. Brasil. Mat. 14 (1983), 27-43. 
  13. [MR] R. Ma né and L. F. da Rocha, Julia sets are uniformly perfect, Proc. Amer. Math. Soc. 116 (1992), 251-257. Zbl0763.30010
  14. [SS] M. Shub and D. Sullivan, Expanding endomorphisms of the circle revisited, Ergodic Theory Dynam. Systems 5 (1985), 285-289. Zbl0583.58022
  15. [Z1] A. Zdunik, Parabolic orbifolds and the dimension of the maximal measure for rational maps, Invent. Mat. 99 (1990), 627-649. Zbl0820.58038
  16. [Z2] A. Zdunik, Invariant measure in the class of harmonic measures for polynomial-like mappings, preprint 538, Inst. Math., Polish Acad. Sci., 1995. 

NotesEmbed ?

top

You must be logged in to post comments.

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

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.