Rigidity of harmonic measure

• Autorzy: I. Popovici , Alexander Volberg
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 30D05: Functional equations in the complex domain, iteration and composition of analytic functions

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 1996
• Tom: 150
• Numer: 3
• Strony: 237-244

Daty

wydano

1996

otrzymano

1995-04-28

poprawiono

1995-12-29

Twórcy

autor

I. Popovici

• Department of Mathematics, Michigan State University, East Lansing, Michigan 48824, U.S.A.

autor

Alexander Volberg

• Department of Mathematics, Michigan State University, East Lansing, Michigan 48824, U.S.A.

Bibliografia

• [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.
• [DH] A. Douady and F. Hubbard, On the dynamics of polynomial like mappings, Ann. Sci. Ecole Norm. Sup. 18 (1985), 287-345.
• [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.
• [SS] M. Shub and D. Sullivan, Expanding endomorphisms of the circle revisited, Ergodic Theory Dynam. Systems 5 (1985), 285-289.
• [Z1] A. Zdunik, Parabolic orbifolds and the dimension of the maximal measure for rational maps, Invent. Mat. 99 (1990), 627-649.
• [Z2] A. Zdunik, Invariant measure in the class of harmonic measures for polynomial-like mappings, preprint 538, Inst. Math., Polish Acad. Sci., 1995.