Ergodicity of conformal measures for unimodal polynomials
Author:
Eduardo A. Prado
Journal:
Conform. Geom. Dyn. 2 (1998), 2944
MSC (1991):
Primary 58F03, 58F23
Published electronically:
March 25, 1998
MathSciNet review:
1613051
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Abstract: Let be a polynomial and a conformal measure for , i.e., a Borel probability measure with Jacobian equal to . We show that if is a real unimodal polynomial (a polynomial with just one critical point), then is ergodic. We also show that is ergodic if is a complex unimodal polynomial with one parabolic periodic point or a quadratic polynomial in the class with a priori bounds (as defined in Lyubich (1997)).
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 A. M. Blokh and M. Lyubich, Measurable dynamics of unimodal maps of the interval, Ann. Sci. École Norm. Sup., 24, 545573, 1991. MR 93f:58132
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 R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, volume 470 of Lecture Notes in Mathematics, SpringerVerlag, 1975. MR 56:1364
 [DH85]
 A. Douady and J. Hubbard, On the dynamics of polynomiallike mappings, Ann. Sci. École Norm. Sup., 18, 287343, 1985. MR 87f:58083
 [DU91a]
 M. Denker and M. Urbanski, On Sullivan's conformal measures for rational maps of the Riemann sphere, Nonlinearity, 4, 365384, (1991). MR 92f:58097
 [DU91b]
 M. Denker and M. Urbanski, Hausdorff and conformal measures on Julia sets with a rationally indifferent periodic point, J. London Math. Soc., 2(43), 107118, 1991. MR 92k:58153
 [Fed69]
 F. Federer, Geometric measure theory, SpringerVerlag, 1969. MR 41:1976
 [GS]
 J. Graczyk and G. Swiatek, Polynomiallike property for real quadratic polynomials, preprint, 1995.
 [Hub]
 J. Hubbard, Local connectivity of Julia sets and bifurcation loci: three theorems of J.C. Yoccoz, Topological methods in modern mathematics, A symposium in honor of John Milnor, Publish or Perish, 467511. MR 94c:58172
 [LvS95]
 G. Levin and S. van Strien, Local connectivity of Julia set of real polynomials, Ann. of Math., to appear.
 [Lyu91]
 M. Lyubich, On the Lebesgue measure of the Julia set of a quadratic polynomial, IMSStony Brook preprint series, (1991/10), 1991.
 [Lyu97]
 M. Lyubich, Dynamics of quadratic polynomials, III, Acta Math., 178, 185297, 1997. CMP 97:15
 [LyuY95]
 M. Lyubich and M. Yampolski, Complex bounds for real polynomials, Ann. Inst. Fourier, 47, 12191255, 1997.
 [McM94]
 C. McMullen, Complex dynamics and renormalization, Number 135, Princeton Univ. Press, 1994. MR 96b:58097
 [McM95]
 C. McMullen, The classification of conformal dynamical systems, preprint, 1995. CMP 98:02
 [MvS93]
 W. de Melo and S. van Strien, One dimensional dynamics, SpringerVerlag, 1993. MR 95a:58035
 [Mil90]
 J. Milnor, Dynamics in one complex variable: Introductory lectures, IMSStony Brook preprint series, (1990/5), 1990.
 [Mil91]
 J. Milnor, Local connectivity of Julia sets: expository lectures, IMSStony Brook preprint series, (1991/10), 1991.
 [Pra95]
 E. A. Prado, Conformal measures in polynomial dynamics, In PhD thesis, SUNY at Stony Brook, 1995.
 [Prz]
 F. Przytycki, Iterations of holomorphic ColletEckmann maps: conformal and invariant measures, preprint, 1996. CMP 96:17
 [Sul80]
 D. Sullivan, Conformal dynamics, 725752, volume 1007 of Lecture Notes in Mathematics, SpringerVerlag, 1980.
 [U]
 M. Urbanski, Rational functions with no recurrent critical points, Ergodic Theory Dynam. Systems 14 (1994), no. 2, 391414. MR 95g:58191
 [Wal78]
 P. Walters, Invariant measures and equilibrium states for some mappings which expand distance, Trans. Amer. Math. Soc., 263, 121153, 1978. MR 57:6371
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Additional Information
Eduardo A. Prado
Affiliation:
Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 66281 CEP 05315970, São Paulo, Brazil
Email:
prado@ime.usp.br
DOI:
http://dx.doi.org/10.1090/S1088417398000198
PII:
S 10884173(98)000198
Keywords:
Holomorphic dynamics,
conformal measures
Received by editor(s):
September 1, 1997
Received by editor(s) in revised form:
December 15, 1997
Published electronically:
March 25, 1998
Additional Notes:
Supported in part by CNPqBrazil and S.U.N.Y. at Stony Brook
Article copyright:
© Copyright 1998 American Mathematical Society
