Poincare seminar 2010: Chaos by Bertrand Duplantier, Stéphane Nonnenmacher, Vincent

Number Theory

By Bertrand Duplantier, Stéphane Nonnenmacher, Vincent Rivasseau

This 12th quantity within the Poincaré Seminar sequence provides an entire and interdisciplinary standpoint at the notion of Chaos, either in classical mechanics in its deterministic model, and in quantum mechanics. This booklet expounds the most vast ranging questions in technological know-how, from uncovering the fingerprints of classical chaotic dynamics in quantum structures, to predicting the destiny of our personal planetary process. Its seven articles also are hugely pedagogical, as befits their foundation in lectures to a wide clinical viewers. Highlights contain an entire description by way of the mathematician É. Ghys of the paradigmatic Lorenz attractor, and of the famed Lorenz butterfly influence because it is known this present day, illuminating the basic mathematical matters at play with deterministic chaos; an in depth account by means of the experimentalist S. Fauve of the masterpiece scan, the von Kármán Sodium or VKS scan, which demonstrated in 2007 the spontaneous new release of a magnetic box in a strongly turbulent circulate, together with its reversal, a version of Earth’s magnetic box; an easy toy version by way of the theorist U. Smilansky – the discrete Laplacian on finite d-regular expander graphs – which permits one to know the basic constituents of quantum chaos, together with its primary hyperlink to random matrix idea; a evaluate through the mathematical physicists P. Bourgade and J.P. Keating, which illuminates the attention-grabbing connection among the distribution of zeros of the Riemann ζ-function and the statistics of eigenvalues of random unitary matrices, which may eventually supply a spectral interpretation for the zeros of the ζ-function, therefore an explanation of the distinguished Riemann speculation itself; a piece of writing via a pioneer of experimental quantum chaos, H-J. Stöckmann, who exhibits intimately how experiments at the propagation of microwaves in 2nd or 3D chaotic cavities fantastically make sure theoretical predictions; an intensive presentation through the mathematical physicist S. Nonnenmacher of the “anatomy” of the eigenmodes of quantized chaotic structures, particularly in their macroscopic localization houses, as governed by means of the Quantum Ergodic theorem, and of the deep mathematical problem posed by way of their fluctuations on the microscopic scale; a evaluate, either ancient and clinical, by means of the astronomer J. Laskar at the balance, consequently the destiny, of the chaotic sunlight planetary procedure we are living in, a subject matter the place he made groundbreaking contributions, together with the probabilistic estimate of attainable planetary collisions. This e-book could be of vast common curiosity to either physicists and mathematicians

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This proposal, controversial at first, has gained much favor. It seems that Smale had very few physical motivations when cooking up his theory of hyperbolic systems, while physics itself does not seem to encompass many hyperbolic systems. This is at least Anosov’s point of view [4]: One gets the impression that the Lord God would prefer to weaken hyperbolicity a bit rather than deal with restrictions on the topology of an attractor that arise when it really is “1960s-model” hyperbolic. Even nowadays, it is not easy to find physical phenomena with strictly hyperbolic dynamics (see however [35, 39]).

Following a nonwandering geodesic on the pants ???? , after each crossing with a seam one may consider to turn right or turn left to reach the next seam. It is thus possible to associate to each nonwandering geodesic a bi-infinite sequence of “left/right” symbols. Yet, this new coding is not perfect, because it is not bijective. Assume the pants is embedded symmetrically in space, meaning that it is invariant 38 E. Ghys Figure 20. Symmetries of the pants through six rotations (the identity, two rotations of order three, and three rotations of order two), as in Figure 20.

Thus within the limits of accuracy of the printed values, the trajectory is confined to a pair of surfaces which appear to merge in the lower portion. [. . ] It would seem, then, that the two surfaces merely appear to merge, and remain distinct surfaces. [. . , and we finally conclude that there is an infinite complex of surfaces, each extremely close to one or the other of the two merging surfaces. Figure 12 is reprinted from Lorenz’s article. Starting from an initial condition, the orbit rapidly approaches this “two-dimensional object” and then travels “on” this surface.

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