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Physical measures for infinitely renormalizable Lorenz maps

Published online by Cambridge University Press:  19 September 2016

M. MARTENS  and
B. WINCKLER
M. MARTENS
Affiliation:
Department of Mathematics, Stony Brook University, Stony Brook, NY 11794-3651, USA
B. WINCKLER
Affiliation:
Department of Mathematics, KTH, 100 44 Stockholm, Sweden email winckler@kth.se
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Abstract

A physical measure on the attractor of a system describes the statistical behavior of typical orbits. An example occurs in unimodal dynamics: namely, all infinitely renormalizable unimodal maps have a physical measure. For Lorenz dynamics, even in the simple case of infinitely renormalizable systems, the existence of physical measures is more delicate. In this article, we construct examples of infinitely renormalizable Lorenz maps which do not have a physical measure. A priori bounds on the geometry play a crucial role in (unimodal) dynamics. There are infinitely renormalizable Lorenz maps which do not have a priori bounds. This phenomenon is related to the position of the critical point of the consecutive renormalizations. The crucial technical ingredient used to obtain these examples without a physical measure is the control of the position of these critical points.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 2 , April 2018 , pp. 717 - 738
Copyright
© Cambridge University Press, 2016 

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References

Chandramouli, V. V. M. S., Martens, M., de Melo, W. and Tresser, C. P.. Chaotic period doubling. Ergod. Th. & Dynam. Sys. 29 (2009), 381–418.Google Scholar
Coullet, P. and Tresser, C.. Itération d’endomorphismes et groupe de renormalisation. J. Phys. Colloq. C 539 (1978), C5–25.Google Scholar
Feigenbaum, M. J.. Quantitative universality for a class of non-linear transformations. J. Stat. Phys. 19 (1978), 25–52.Google Scholar
Gambaudo, J.-M. and Martens, M.. Algebraic topology for minimal cantor sets. Ann. Henri Poincaré 7(3) (2006), 423–446.Google Scholar
Hofbauer, F. and Keller, G.. Quadratic maps without asymptotic measure. Comm. Math. Phys. 127 (1990), 319–337.Google Scholar
Johnson, S.. Singular measures without restrictive intervals. Comm. Math. Phys. 110 (1987), 185–190.Google Scholar
Lorenz, E. N.. Deterministic non-periodic flow. J. Atmos. Sci. 20 (1963), 130–141.2.0.CO;2>CrossRefGoogle Scholar
Martens, M.. Distortion results and invariant cantor sets for unimodal maps. Ergod. Th. & Dynam. Sys. 14 (1994), 331–349.Google Scholar
Martens, M.. The periodic points of renormalization. Ann. of Math. (2) 147 (1998), 543–584.Google Scholar
Misiurewicz, M.. Absolutely continuous measures for certain maps of an interval. Publ. Math. Inst. Hautes Études Sci. 53(1) (1981), 17–51.Google Scholar
Martens, M. and de Melo, W.. Universal models for Lorenz maps. Ergod. Th. & Dynam. Sys. 21(3) (2001), 833–860.Google Scholar
de Melo, W. and van Strien, S.. One-dimensional Dynamics. Springer, Berlin, 1993.Google Scholar
Martens, M. and Winckler, B.. On the hyperbolicity of Lorenz renormalization. Comm. Math. Phys. 325(1) (2013), 185–257.Google Scholar
Tucker, W.. The Lorenz attractor exists. C. R. Math. Acad. Sci. Paris Sér I 328.12 (1999), 1197–1202.Google Scholar
Viana, M.. What’s new on Lorenz strange attractors? Math. Intelligencer 22(3) (2000), 6–19.Google Scholar
Winckler, B.. Renormalization of Lorenz Maps. PhD Thesis KTH, Stockholm, Sweden, 2011.Google Scholar