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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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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