By P. Collet (auth.), P. Collet, M. Courbage, S. Métens, A. Neishtadt, G. Zaslavsky (eds.)
From the 18th to the thirtieth August 2003 , a NATO complex examine Institute (ASI) used to be held in Cargèse, Corsica, France. Cargèse is a pleasant small village positioned via the mediterranean sea and the Institut d'Etudes Scientifiques de Cargese offers ? a standard position to arrange Theoretical Physics summer time faculties and Workshops * in a closed and good equiped position. The ASI was once a global summer season university on "Chaotic Dynamics and shipping in Classical and Quantum Systems". the most target of the varsity used to be to enhance the mutual interplay among Physics and arithmetic touching on statistical homes of classical and quantum dynamical structures. numerous experimental and numerical observations have proven new phenomena of chaotic and anomalous delivery, fractal buildings, chaos in physics accelerators and in cooled atoms inside of atom-optics billiards, space-time chaos, fluctuations faraway from equilibrium, quantum decoherence and so forth. New theoretical equipment were constructed in an effort to modelize and to appreciate those phenomena (volume keeping and ergodic dynamical platforms, non-equilibrium statistical dynamics, fractional kinetics, coupled maps, space-time entropy, quantum dissipative approaches etc). the varsity collected a group of experts from numerous horizons lecturing and discussing at the achievements, views and open difficulties (both basic and applied).
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Extra resources for Chaotic Dynamics and Transport in Classical and Quantum Systems: Proceedings of the NATO Advanced Study Institute on International Summer School on Chaotic Dynamics and Transport in Classical and Quantum Systems Cargèse, Corsica 18–30 August 2003
Collet in this volume ). Hopf, Koopman and von Neumann studied the relationship between mixing properties and the spectrum of U showing that for T , being weakly mixing is equivalent, for U , to having continuous spectrum [16, 19]. Later on, von Neumann and Halmos classiﬁed completely the dynamical systems having purely discrete spectrum. All these results are presented in the Halmos book in 1956 . In the sixties, many progresses were made in studying the stucture of the continuous spectrum of K-systems with works of Kolmogorov, Sinai and Rokhlin .
We call 22 Figure 1. Trajectory of particle in the plate billiard with slits n the ”veolcity direction of the particle”. 13) So T maps an ingoing arrow to the next ingoing arrow. When G is endowed with the Haar measure m(±1) = 1/2, T is a two-elements group extension of the rotation. Spectral Properties As well known, the space H = L2 (S 1 × G, µ × m) is a tensor product L2 (S 1 , µ) ⊗ L2 (G, m). 15) In other words, for any f ∈ L2 (S 1 , µ) ⊗ L2 (G, m), there are g, h ∈ L2 (S 1 , µ) such that : f (x, ε) = g(x) + εh(x).
In the case of subshifts. In these cases Hausdorﬀ dimensions of invariant sets can be expressed in terms of topological pressure. It was R. Bowen who introduced this quantity in the theory of dynamical systems [34, 20]. 2. Dynamical Chaos We give a deﬁnition of dynamical chaos following mainly Takens’ ideas see [39, 40] and . Dynamical Chaos in Terms of the -complexity 41 We say that a signal (an observable) φ(t) is generated by a dynamical system (f t , M ) if there exist an initial point x0 belonging to M and the function ψ : M → R such that φ(t) = ψ(f t x0 ).
Chaotic Dynamics and Transport in Classical and Quantum Systems: Proceedings of the NATO Advanced Study Institute on International Summer School on Chaotic Dynamics and Transport in Classical and Quantum Systems Cargèse, Corsica 18–30 August 2003 by P. Collet (auth.), P. Collet, M. Courbage, S. Métens, A. Neishtadt, G. Zaslavsky (eds.)