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Circle diffeomorphisms forced by expanding circle maps

Published online by Cambridge University Press:  01 November 2011

ALE JAN HOMBURG
ALE JAN HOMBURG*
Affiliation:
KdV Institute for Mathematics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands Department of Mathematics, VU University Amsterdam, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands (email: a.j.homburg@uva.nl)
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Abstract

We discuss the dynamics of skew product maps defined by circle diffeomorphisms forced by expanding circle maps. We construct an open class of such systems that are robustly topologically mixing and for which almost all points in the same fiber converge under iteration. This property follows from the construction of an invariant attracting graph in the natural extension, a skew product of circle diffeomorphisms forced by a solenoid homeomorphism.

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Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 6 , December 2012 , pp. 2011 - 2024
Copyright
Copyright © Cambridge University Press 2011

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References

[1]Antonov, V. A.. Modeling of processes of cyclic evolution type: synchronization by a random signal. Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 2 (1984), 67–76.Google Scholar
[2]Arnold, L.. Random Dynamical Systems (Springer Monographs in Mathematics). Springer, Berlin, 1998.Google Scholar
[3]Avila, A. and Viana, M.. Extremal Lyapunov exponents: an invariance principle and applications. Invent. Math. 181 (2010), 115–174.CrossRefGoogle Scholar
[4]Boccaletti, S., Kurths, J., Osipov, G., Valladares, D. L. and Zhou, C. S.. The synchronization of chaotic systems. Phys. Rep. 366(1–2) (2002), 1–101.Google Scholar
[5]Bonatti, C. and Díaz, L. J.. Persistent nonhyperbolic transitive diffeomorphisms. Ann. of Math. (2) 143(2) (1996), 357–396.CrossRefGoogle Scholar
[6]Bonatti, C., Díaz, L. J. and Gorodetskiĭ, A. S.. Non-hyperbolic ergodic measures with large support. Nonlinearity 23(3) (2010), 687–706.Google Scholar
[7]Bonatti, C., Díaz, L. J. and Viana, M.. Dynamics Beyond Uniform Hyperbolicity (Encyclopaedia of Mathematical Sciences, 102). Springer, Berlin, 2005.Google Scholar
[8]Bonatti, C. and Viana, M.. SRB measures for partially hyperbolic systems whose central direction is mostly contracting. Israel J. Math. 115 (2000), 157–193.CrossRefGoogle Scholar
[9]Burns, K., Dolgopyat, D., Pesin, Ya. and Pollicott, M.. Stable ergodicity for partially hyperbolic attractors with negative central exponents. J. Mod. Dyn. 2(1) (2008), 63–81.Google Scholar
[10]Crauel, H.. Extremal exponents of random dynamical systems do not vanish. J. Dynam. Differential Equations 2 (1990), 245–291.CrossRefGoogle Scholar
[11]Deroin, B., Kleptsyn, V. A. and Navas, A.. Sur la dynamique unidimensionnelle en régularité intermédiaire. Acta Math. 199(2) (2007), 199–262.CrossRefGoogle Scholar
[12]Díaz, L. J. and Gorodetskiĭ, A. S.. Non-hyperbolic ergodic measures for non-hyperbolic homoclinic classes. Ergod. Th. & Dynam. Sys. 29(5) (2009), 1479–1513.CrossRefGoogle Scholar
[13]Ghane, G. H., Homburg, A. J. and Sarizadeh, A.. C 1 robustly minimal iterated function systems. Stoch. Dyn. 10(1) (2010), 155–160.CrossRefGoogle Scholar
[14]Gorodetskiĭ, A. S. and Il’yashenko, Yu. S.. Certain properties of skew products over a horseshoe and a solenoid. Dynamical Systems, Automata, and Infinite Groups (Proceedings of the Steklov Institute of Mathematics, 231). Ed. Grigorchuk, R. I.. MAIK Nauka/Interperiod. Publ., Moscow, 2000, pp. 90–112.Google Scholar
[15]Gorodetskiĭ, A. S., Il’yashenko, Yu. S., Kleptsyn, V. A. and Nal’skiĭ, M. B.. Nonremovability of zero Lyapunov exponents. Funct. Anal. Appl. 39(1) (2005), 21–30.Google Scholar
[16]Hirsch, M. W., Pugh, C. C. and Shub, M.. Invariant Manifolds (Lecture Notes in Mathematics, 583). Springer, Berlin, 1977.Google Scholar
[17]Homburg, A. J.. Atomic disintegrations for partially hyperbolic diffeomorphisms. Preprint, 2011, available at http://www.science.uva.nl/∼alejan/publications.html.Google Scholar
[18]Jäger, T. H.. Strange non-chaotic attractors in quasiperiodically forced circle maps. Comm. Math. Phys. 289(1) (2009), 253–289.CrossRefGoogle Scholar
[19]Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems, with a Supplement by Anatole Katok and Leonardo Mendoza (Encyclopedia of Mathematics and Its Applications, 54). Cambridge University Press, Cambridge, 1997.Google Scholar
[20]Kleptsyn, V. A. and Nalskii, M. B.. Contraction of orbits in random dynamical systems on the circle. Funct. Anal. Appl. 38(4) (2004), 267–282.CrossRefGoogle Scholar
[21]Le Jan, Y.. Equilibre statistique pour les produits de difféomorphismes aléatoires indépendants. Ann. Inst. Henri. Poincaré Probab. Stat. 23 (1987), 111–120.Google Scholar
[22]Mañé, R.. Ergodic Theory and Differentiable Dynamics (Ergebnisse der Mathematik und ihrer Grenzgebiete 3 Folge, 8). Springer, Berlin, 1987.Google Scholar
[23]Navas, A.. Groups of Circle Diffeomorphisms. University of Chicago Press, Chicago, IL, 2011.Google Scholar
[24]Palis, J. and Takens, F.. Hyperbolicity & Sensitive Chaotic Dynamics at Homoclinic Bifurcations. Cambridge University Press, Cambridge, 1993.Google Scholar
[25]Pesin, Ya. B. and Sinaĭ, Ya. G.. Gibbs measures for partially hyperbolic attractors. Ergod. Th. & Dynam. Sys. 2(3–4) (1983), 417–438, 1982.Google Scholar
[26]Pesin, Ya. B.. Lectures on Partial Hyperbolicity and Stable Ergodicity (Zurich Lectures in Advanced Mathematics). European Mathematical Society, Zürich, 2004.Google Scholar
[27]Pugh, C., Shub, M. and Wilkinson, A.. Hölder foliations. Duke Math. J. 86(3) (1997), 517–546, Correction. Duke Math. J. 105(1) (2000) 105–106.CrossRefGoogle Scholar
[28]Ruelle, D.. Perturbation theory for Lyapunov exponents of a toral map: extension of a result of Shub and Wilkinson. Israel J. Math. 134 (2003), 345–361.Google Scholar
[29]Ruelle, D. and Wilkinson, A.. Absolutely singular dynamical foliations. Comm. Math. Phys. 219(3) (2001), 481–487.Google Scholar
[30]Shub, M.. Endomorphisms of compact differentiable manifolds. Amer. J. Math. 91(1) (1969), 175–199.CrossRefGoogle Scholar
[31]Shub, M. and Wilkinson, A.. Pathological foliations and removable zero exponents. Invent. Math. 139(3) (2000), 495–508.CrossRefGoogle Scholar
[32]Tsujii, M.. Fat solenoidal attractors. Nonlinearity 14(5) (2001), 1011–1027.CrossRefGoogle Scholar
[33]Tsujii, M.. Physical measures for partially hyperbolic surface endomorphisms. Acta Math. 194(1) (2005), 37–132.Google Scholar
[34]Williams, R. F.. Expanding attractors. Publ. Math. Inst. Hautes Études Sci. 43 (1974), 169–203.Google Scholar
[35]Zmarrou, H. and Homburg, A. J.. Dynamics and bifurcations of random circle diffeomorphisms. Discrete Contin. Dyn. Syst. Ser. B 10(2–3) (2008), 719–731.Google Scholar