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Hyperbolization of cocycles by isometries of the Euclidean space

Published online by Cambridge University Press:  05 December 2012

MARIO PONCE
MARIO PONCE*
Affiliation:
Facultad de Matemáticas, Universidad Católica de Chile, Casilla 306, Santiago 22, Chile (email: [email protected])
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Abstract

We study hyperbolized versions of cohomological equations that appear with cocycles by isometries of the Euclidean space. These (hyperbolized versions of) equations have a unique continuous solution. We concentrate on the question whether or not these solutions converge to a genuine solution to the original equation, and in what sense we can use them as good approximative solutions. The main advantage of considering solutions to hyperbolized cohomological equations is that they can be easily described, since they are global attractors of a naturally defined skew-product dynamics. We also include some technical results about twisted Birkhoff sums and exponential averaging.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 2 , April 2014 , pp. 628 - 646
Copyright
©2012 Cambridge University Press 

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References

[1]Bochi, J. and Navas, A.. A geometric path from zero Lyapunov exponents to rotation cocycles. Preprint, arXiv:1101.3523v4.Google Scholar
[2]Bousch, T.. La condition de Walters. Ann. Sci. Éc. Norm. Supér. 34 (2001), 287–311.Google Scholar
[3]Boyarsky, A. and Góra, P.. Laws of chaos. Invariant Measures and Dynamical Systems in One Dimension (Probability and its Applications). Birkhäuser Boston, Boston, 1997.Google Scholar
[4]Coronel, D., Navas, A. and Ponce, M.. On bounded cocycles of isometries over a minimal dynamics. Preprint, arXiv:1101.3523v4.Google Scholar
[5]Coronel, D., Navas, A. and Ponce, M.. On the dynamics of non-reducible cylindrical vortices. J. Lond. Math. Soc. 85(3) (2012), 789–808.Google Scholar
[6]de la Llave, R. and Windsor, A.. Livšic theorems for non-commutative groups including diffeomorphism groups and results on the existence of conformal structures for Anosov systems. Ergod. Th. & Dynam. Syst. 30(4) (2010), 1055–1110.Google Scholar
[7]Gottschalk, W. H. and Hedlund, G. A.. Topological Dynamics. American Mathematical Society, Providence, RI, 1955.Google Scholar
[8]Hardy, G. H.. Divergent Series, with a preface by J. E. Littlewood and a note by L. S. Bosanquet. Reprint of the revised (1963) edition. Éditions Jacques Gabay, Sceaux, 1992.Google Scholar
[9]Jenkinson, O.. Ergodic optimization. Discrete Contin. Dyn. Syst. 15(1) (2006), 197–224.Google Scholar
[10]Jorba, À. and Tatjer, J. C.. A mechanism for the fractalization of invariant curves in quasi-periodically forced 1-D maps. Discrete Contin. Dyn. Syst. Ser. B 10(2–3) (2008), 537–567.Google Scholar
[11]Kalinin, B.. Livšic theorem for matrix cocycles. Ann. of Math. 173(2) (2011), 1025–1042.CrossRefGoogle Scholar
[12]Katok, A.. Cocycles, cohomology and combinatorial constructions in ergodic theory. In collaboration with E. A. Robinson. Jr. Smooth Ergodic Theory and its Applications (Proceedings of Symposia in Pure Mathematics, 69) (Seattle, WA, 1999). American Mathematical Society, Providence, RI, 2001, pp. 107–173.Google Scholar
[13]Katok, A. and Kononenko, A.. Cocycles’ stability for partially hyperbolic systems. Math. Res. Lett. 3(2) (1996), 191–210.Google Scholar
[14]Livšic, A. N.. Cohomology of dynamical systems. Math. USSR Izvestija 6 (1972), 1278–1301.Google Scholar
[15]Petersen, G. M.. Regular Matrix Transformations. McGraw-Hill, London, 1966.Google Scholar
[16]Wilkinson, A.. The cohomological equation for partially hyperbolic diffeomorphisms. Astérisque, to appear, arXiv:0809.4862v2.Google Scholar
[17]Yoccoz, J.-Ch.. Théorème de Siegel, nombres de Bruno et polynômes quadratiques. Petits diviseurs en dimension 1. Astérisque 231 (1995), 3–88.Google Scholar