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Dynamics of two-dimensional Blaschke products

Published online by Cambridge University Press:  01 April 2008

ENRIQUE R. PUJALS  and
MICHAEL SHUB
ENRIQUE R. PUJALS
Affiliation:
IMPA Estrada Dona Castorina 110, Rio de Janeiro, Brasil 22460-320 (email: [email protected])
MICHAEL SHUB
Affiliation:
Math Department, University of Toronto, 40 St. George Street, Toronto, ON M5S 2E4, Canada (email: [email protected])
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Abstract

In this paper we study the dynamics on and of a two-dimensional Blaschke product. We prove that in the case when the Blaschke product is a diffeomorphism of with all periodic points hyperbolic then the dynamics is hyperbolic. If a two-dimensional Blaschke product diffeomorphism of is embedded in a two-dimensional family given by composition with translations of , then we show that there is a non-empty open set of parameter values for which the dynamics is Anosov or has an expanding attractor with a unique SRB measure.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 2: William Parry Memorial Volume , April 2008 , pp. 575 - 585
Copyright
Copyright © Cambridge University Press 2008

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References

[1]Bonatti, C., Diaz, L. J. and Viana, M.. Dynamics Beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective. Springer, 2005.Google Scholar
[2]Carleson, L. and Gamelin, T.. Complex Dynamics (Universitext: Tracts in Mathematics). Springer, 1993.CrossRefGoogle Scholar
[3]Franks, J.. Anosov diffeomorphisms. Global Analysis (Proceedings of Symposia in Pure Mathematics, XIV). American Mathematical Society, Providence, RI, 1970.Google Scholar
[4]Manning, A.. There are no new Anosov diffeomorphisms on tori. Amer. J. Math. 96 (1974), 422–429.CrossRefGoogle Scholar
[5]Pujals, E. R., Robert, L. and Shub, M.. Expanding maps of the circle rerevisited: positive Lyapunov exponents in a rich family. Ergod. Th. & Dynam. Sys. 26(6) (2006), 1931–1937.CrossRefGoogle Scholar
[6]Pujals, E. R. and Sambarino, M.. Homoclinic tangencies and hyperbolicity for surface diffeomorphisms. Ann. of Math. 151 (2000), 961–1023.CrossRefGoogle Scholar
[7]Pujals, E. R. and Sambarino, M.. On the dynamics of dominated splitting. Ann. of Math., to appear.Google Scholar
[8]Rudin, W.. Function Theory in Polydiscs. W.A. Benjamin, New York, 1969.Google Scholar
[9]Shub, M. and Sullivan, D.. A remark on the Lefschetz fixed point formula for differentiable maps. Topology 13 (1974), 189–191.CrossRefGoogle Scholar