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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 [email protected]
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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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