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 research Institute (ASI) was once held in Cargèse, Corsica, France. Cargèse is a pleasant small village located by means of the mediterranean sea and the Institut d'Etudes Scientifiques de Cargese presents ? a standard position to prepare Theoretical Physics summer season colleges and Workshops * in a closed and good equiped position. The ASI was once a world summer season college on "Chaotic Dynamics and shipping in Classical and Quantum Systems". the most target of the varsity used to be to increase the mutual interplay among Physics and arithmetic pertaining to statistical houses of classical and quantum dynamical platforms. a variety of experimental and numerical observations have proven new phenomena of chaotic and anomalous shipping, fractal constructions, chaos in physics accelerators and in cooled atoms inside of atom-optics billiards, space-time chaos, fluctuations faraway from equilibrium, quantum decoherence and so on. New theoretical equipment were constructed with the intention to modelize and to appreciate those phenomena (volume retaining and ergodic dynamical platforms, non-equilibrium statistical dynamics, fractional kinetics, coupled maps, space-time entropy, quantum dissipative approaches etc). the college amassed a staff of experts from a number of horizons lecturing and discussing at the achievements, views and open difficulties (both basic and applied).

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**Extra info 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**

**Example text**

The shift map T a = (a1 , a2 , . ) generates a “universal” dynamical system that generates all bounded observables. i Fix an observable a = (a0 , a1 , . ) and denote by M the set clos ∞ i=0 T a , the set of limiting points of the orbit {T i a }. We say that an observable a = ( a0 , a1 , . . ) is deterministically generated if 1 The fractal dimension dimB M = b < ∞ (16) 2 The topological entropy of f = T |M , h(f ) < ∞. (17) Because of the Ma˜ ne Theorem (see for instance[8]), the condition 16 implies that, generally speaking, for any n ≥ 2b + 1 the system (f t , M ) is topologically conjugated to the system generated by the map Tˆ : (ak , .

Qi −1} at every coordinate i = 0, 1, . . Such systems will be referred to as polysymbolic systems. The set of sequences is Ωq∗ = A0 × A1 × · · · , and the size of alphabets at every coordinate i is denoted by the sequence q∗ = (q0 , q1 , . ) of positive integers qi = card(Ai ), i ≥ 0. A polysymbolic system (Ωq∗ , T ) is multipermutative if for every ω = (ω0 , ω1 , . . ) the map T : Ωq∗ → Ωq∗ is deﬁned by T ω = (ω0 + p0 , ω1 + p1 (ω0 ), . . , ωi + pi (ω0 , . . , ωi−1 ), . . ) with pi : A0 × · · · × Ai−1 → Ai and p0 ∈ A0 .

19 Notes on Spectral Theory, Mixing and Transport. Singular continuous measures σf are such that one can ﬁnd a set of zero Lebesgue measure which is of measure 1 for σf and vice versa. They are often obtained in DS with zero entropy. e. 5 Let (X, T, µ) be a DS. • T is mixing iﬀ, for any function f such that f dµ = 0, σˆf (n) → 0 as n → ∞ • T is weakly mixing iﬀ for any function f such that f dµ = 0, σf is continuous. Rapid decay of correlations is the short memory signature of deterministic dynamical systems, among them the class of Bernoulli systems is the most ”stochastic”.