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Physical measures for infinitely renormalizable Lorenz maps
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2018 (English)In: Ergodic Theory and Dynamical Systems, ISSN 0143-3857, E-ISSN 1469-4417, Vol. 38, p. 717-738Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
Cambridge University Press, 2018. Vol. 38, p. 717-738
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-225171DOI: 10.1017/etds.2016.43ISI: 000427509200013OAI: oai:DiVA.org:kth-225171DiVA, id: diva2:1194990
Funder
Knut and Alice Wallenberg Foundation
Note

QC 20180404

Available from: 2018-04-04 Created: 2018-04-04 Last updated: 2018-04-11Bibliographically approved

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