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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Analyticity of the SRB measure for holomorphic families of quadratic-like Collet-Eckmann maps


Authors: Viviane Baladi and Daniel Smania
Journal: Proc. Amer. Math. Soc. 137 (2009), 1431-1437
MSC (2000): Primary 37C40, 37C30, 37D25, 37E05
Published electronically: October 27, 2008
MathSciNet review: 2465669
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Abstract: We show that if $ f_t$ is a holomorphic family of quadratic-like maps with all periodic orbits repelling so that for each real $ t$ the map $ f_t$ is a real Collet-Eckmann $ S$-unimodal map, then, writing $ \mu_t$ for the unique absolutely continuous invariant probability measure of $ f_t$, the map

$\displaystyle t\mapsto \int \psi \, d\mu_t $

is real analytic for any real analytic function $ \psi$.


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Additional Information

Viviane Baladi
Affiliation: Département de Mathématiques et Applications, UMR 8553, École Normale Supérieure, 75005 Paris, France
Email: viviane.baladi@ens.fr

Daniel Smania
Affiliation: Departamento de Matemática, ICMC-USP, Caixa Postal 668, São Carlos-SP, CEP 13560-970 São Carlos-SP, Brazil
Email: smania@icmc.usp.br

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09651-2
PII: S 0002-9939(08)09651-2
Received by editor(s): January 22, 2008
Received by editor(s) in revised form: May 27, 2008
Published electronically: October 27, 2008
Additional Notes: The first author is partially supported by ANR-05-JCJC-0107-01. She wrote part of this paper while visiting the Universidad Católica del Norte, Antofagasta, Chile, whose hospitality is gratefully acknowledged. We thank D. Sands for very helpful comments.
The second author is partially supported by CNPq 470957/2006-9 and 310964/2006-7, FAPESP 2003/03107-9. He thanks the DMA of École Normale Supérieure for hospitality during a visit where a crucial part of this work was done.
Communicated by: Jane M. Hawkins
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.