Skew product attractors and concavity

Alsedà, Lluís; Misiurewicz, Michał

doi:10.1090/S0002-9939-2014-12271-4

Skew product attractors and concavity
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by Lluís Alsedà and Michał Misiurewicz

Proc. Amer. Math. Soc. 143 (2015), 703-716

DOI: https://doi.org/10.1090/S0002-9939-2014-12271-4

Published electronically: October 3, 2014

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Abstract:

We propose an approach to the attractors of skew products that tries to avoid unnecessary structures on the base space and rejects the assumption on the invariance of an attractor. When nonivertible maps in the base are allowed, one can encounter the mystery of the vanishing attractor. In the second part of the paper, we show that if the fiber maps are concave interval maps, then contraction in the fibers does not depend on the map in the base.

References

Lluís Alsedà and MichałMisiurewicz, Attractors for unimodal quasiperiodically forced maps, J. Difference Equ. Appl. 14 (2008), no. 10-11, 1175–1196. MR 2447193, DOI 10.1080/10236190802332274
Ll. Alsedà and M. Misiurewicz, Random interval homeomorphisms, preprint, 2013.
Ludwig Arnold, Random dynamical systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR 1723992, DOI 10.1007/978-3-662-12878-7
Kristian Bjerklöv, Positive Lyapunov exponent and minimality for a class of one-dimensional quasi-periodic Schrödinger equations, Ergodic Theory Dynam. Systems 25 (2005), no. 4, 1015–1045. MR 2158395, DOI 10.1017/S0143385704000999
Kristian Bjerklöv, Dynamics of the quasi-periodic Schrödinger cocycle at the lowest energy in the spectrum, Comm. Math. Phys. 272 (2007), no. 2, 397–442. MR 2300248, DOI 10.1007/s00220-007-0238-y
Kristian Bjerklöv, SNA’s in the quasi-periodic quadratic family, Comm. Math. Phys. 286 (2009), no. 1, 137–161. MR 2470927, DOI 10.1007/s00220-008-0626-y
Hans Crauel and Franco Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields 100 (1994), no. 3, 365–393. MR 1305587, DOI 10.1007/BF01193705
Àlex Haro, On strange attractors in a class of pinched skew products, Discrete Contin. Dyn. Syst. 32 (2012), no. 2, 605–617. MR 2837075, DOI 10.3934/dcds.2012.32.605
Tobias Jäger, Strange non-chaotic attractors in quasiperiodically forced circle maps, Comm. Math. Phys. 289 (2009), no. 1, 253–289. MR 2504850, DOI 10.1007/s00220-009-0753-0
Àngel Jorba, Joan Carles Tatjer, Carmen Núñez, and Rafael Obaya, Old and new results on strange nonchaotic attractors, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 17 (2007), no. 11, 3895–3928. MR 2384391, DOI 10.1142/S0218127407019780
Gerhard Keller, A note on strange nonchaotic attractors, Fund. Math. 151 (1996), no. 2, 139–148. MR 1418993
Sergiĭ Kolyada, MichałMisiurewicz, and Ľubomír Snoha, Topological entropy of nonautonomous piecewise monotone dynamical systems on the interval, Fund. Math. 160 (1999), no. 2, 161–181. MR 1706474, DOI 10.4064/fm-160-2-161-181
John Milnor, On the concept of attractor, Comm. Math. Phys. 99 (1985), no. 2, 177–195. MR 790735
William Parry, Entropy and generators in ergodic theory, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0262464
Michael Scheutzow, Comparison of various concepts of a random attractor: a case study, Arch. Math. (Basel) 78 (2002), no. 3, 233–240. MR 1888707, DOI 10.1007/s00013-002-8241-1