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Conformal Geometry and Dynamics

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Ergodicity of conformal measures for unimodal polynomials

Author: Eduardo A. Prado
Journal: Conform. Geom. Dyn. 2 (1998), 29-44
MSC (1991): Primary 58F03, 58F23
Published electronically: March 25, 1998
MathSciNet review: 1613051
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Abstract: Let $f$ be a polynomial and $\mu$ a conformal measure for $f$, i.e., a Borel probability measure $\mu$ with Jacobian equal to $|Df(z)|^{\delta}$. We show that if $f$ is a real unimodal polynomial (a polynomial with just one critical point), then $\mu$ is ergodic. We also show that $\mu$ is ergodic if $f$ is a complex unimodal polynomial with one parabolic periodic point or a quadratic polynomial in the $\mathcal{SL}$ class with a priori bounds (as defined in Lyubich (1997)).

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  • [BL91] A. M. Blokh and M. Yu. Lyubich, Measurable dynamics of 𝑆-unimodal maps of the interval, Ann. Sci. École Norm. Sup. (4) 24 (1991), no. 5, 545–573. MR 1132757
  • [Bow75] Rufus Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975. MR 0442989
  • [DH85] Adrien Douady and John Hamal Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 2, 287–343. MR 816367
  • [DU91a] M. Denker and M. Urbański, On Sullivan’s conformal measures for rational maps of the Riemann sphere, Nonlinearity 4 (1991), no. 2, 365–384. MR 1107011
  • [DU91b] M. Denker and M. Urbański, Hausdorff and conformal measures on Julia sets with a rationally indifferent periodic point, J. London Math. Soc. (2) 43 (1991), no. 1, 107–118. MR 1099090, 10.1112/jlms/s2-43.1.107
  • [Fed69] Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR 0257325
  • [GS] J. Graczyk and G. Swiatek, Polynomial-like property for real quadratic polynomials, preprint, 1995.
  • [Hub] J. H. Hubbard, Local connectivity of Julia sets and bifurcation loci: three theorems of J.-C. Yoccoz, Topological methods in modern mathematics (Stony Brook, NY, 1991) Publish or Perish, Houston, TX, 1993, pp. 467–511. MR 1215974
  • [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, IMS-Stony Brook preprint series, (1991/10), 1991.
  • [Lyu97] M. Lyubich, Dynamics of quadratic polynomials, I-II, Acta Math., 178, 185-297, 1997. CMP 97:15
  • [LyuY95] M. Lyubich and M. Yampolski, Complex bounds for real polynomials, Ann. Inst. Fourier, 47, 1219-1255, 1997.
  • [McM94] Curtis T. McMullen, Complex dynamics and renormalization, Annals of Mathematics Studies, vol. 135, Princeton University Press, Princeton, NJ, 1994. MR 1312365
  • [McM95] C. McMullen, The classification of conformal dynamical systems, preprint, 1995. CMP 98:02
  • [MvS93] Welington de Melo and Sebastian van Strien, One-dimensional dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 25, Springer-Verlag, Berlin, 1993. MR 1239171
  • [Mil90] J. Milnor, Dynamics in one complex variable: Introductory lectures, IMS-Stony Brook preprint series, (1990/5), 1990.
  • [Mil91] J. Milnor, Local connectivity of Julia sets: expository lectures, IMS-Stony 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 Collet-Eckmann maps: conformal and invariant measures, preprint, 1996. CMP 96:17
  • [Sul80] D. Sullivan, Conformal dynamics, 725-752, volume 1007 of Lecture Notes in Mathematics, Springer-Verlag, 1980.
  • [U] Mariusz Urbański, Rational functions with no recurrent critical points, Ergodic Theory Dynam. Systems 14 (1994), no. 2, 391–414. MR 1279476, 10.1017/S0143385700007926
  • [Wal78] Peter Walters, Invariant measures and equilibrium states for some mappings which expand distances, Trans. Amer. Math. Soc. 236 (1978), 121–153. MR 0466493, 10.1090/S0002-9947-1978-0466493-1

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Additional Information

Eduardo A. Prado
Affiliation: Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 66281 CEP 05315-970, São Paulo, Brazil

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 CNPq-Brazil and S.U.N.Y. at Stony Brook
Article copyright: © Copyright 1998 American Mathematical Society