Analyticity of the SRB measure for holomorphic families of quadratic-like Collet-Eckmann maps
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- by Viviane Baladi and Daniel Smania
- Proc. Amer. Math. Soc. 137 (2009), 1431-1437
- DOI: https://doi.org/10.1090/S0002-9939-08-09651-2
- Published electronically: October 27, 2008
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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 \[ t\mapsto \int \psi d\mu _t \] is real analytic for any real analytic function $\psi$.References
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Bibliographic Information
- Viviane Baladi
- Affiliation: Département de Mathématiques et Applications, UMR 8553, École Normale Supérieure, 75005 Paris, France
- MR Author ID: 29810
- 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
- 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
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 137 (2009), 1431-1437
- MSC (2000): Primary 37C40, 37C30, 37D25, 37E05
- DOI: https://doi.org/10.1090/S0002-9939-08-09651-2
- MathSciNet review: 2465669