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SRB-measures for coupled map lattices. (English) Zbl 1115.37006
Chazottes, Jean-René (ed.) et al., Dynamics of coupled map lattices and of related spatially extended systems. Lectures delivered at the school-forum CML 2004, Paris, France, June 21 – July 2, 2004. Berlin: Springer (ISBN 3-540-24289-9/hbk). Lecture Notes in Physics 671, 95-114 (2005).
The article is split into three subsections. The first presents a review of some of the main existence results regarding SRB-measures for coupled map lattices. The main content of this first section is an overview of the proof of a theorem due to Bricmont and Kupiainen who established not only the existence of SRB-measures on the space \(\Pi_{\mathbb{Z}^d} S^1\) where the local dynamics are given by orientation preserving \(C^{1+\delta}\)-maps, but also that the constructed measure various desirable mixing properties. The discussion outlines the three main tools used in the proof; the Perron-Frobenius operator, equilibrium states associated to interactions in lattice gas models and polymer expansions. The first section ends with a brief discussion of the conjecture of Bricmont and Kupiainen that the constructed measure is the only such SRB-measure in their particular set-up. This is a precursor for the third section, where the author constructs a counterexample to this conjecture.
The second section contains a brief overview of some of the main results in geometric measure theory relating the preservation of Hausdorff dimension under projections. Namely that a typical projection with respect to the Haar measure on the Grassmann manifold of all \(m\)-planes where \(m\) is strictly less than the ambient space \(\mathbb{R}^n\) preserves the Hausdorff dimension of a set. The results discussed form the back bone of the ideas used in the proceeding section.
The final section of the paper provides a counterexample to the conjecture of Bricmont and Kupiainen. Using the ideas of the section and constructing a suitable measure supported on the set \(\Pi_{\mathbb{Z}^d} K\), where \(K\) is the standard middle-third Cantor set with associated Hausdorff \(\log2/\log3\)-measure, the author shows that there not only do exist SRB-measures not equal to that provided by the construction of Bricmont-Kupiainen, but there are in fact infinitely many such measures. This is achieved by showing that the coordinate axes in the coupled lattice can be made typical using a slight perturbation operator.
The paper ends with a short discussion of the definiton of SRB-measures and some open problems in the field.
For the entire collection see [Zbl 1073.70002].

37A25 Ergodicity, mixing, rates of mixing
28A78 Hausdorff and packing measures
28A75 Length, area, volume, other geometric measure theory