Ergodicity of conformal measures for unimodal polynomials

Author:
Eduardo A. Prado

Journal:
Conform. Geom. Dyn. **2** (1998), 29-44

MSC (1991):
Primary 58F03, 58F23

DOI:
https://doi.org/10.1090/S1088-4173-98-00019-8

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

Email:
prado@ime.usp.br

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