By Pierre Collet, M. Courbage, S. Métens, A. Neishtadt, G. Zaslavsky

This booklet deals a contemporary up-to-date evaluate at the most crucial actions in this day dynamical platforms and statisitical mechanics by way of the very best specialists within the area. It supplies a modern and pedagogical view on theories of classical and quantum chaos and complexity in hamiltonian and ergodic platforms and their purposes to anomalous delivery in fluids, plasmas, oceans and atom-optic units and to regulate of chaotic shipping. The publication is issued from lecture notes of the overseas summer season university on

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**Example text**

In−1 which satisfy the following assumptions: (A). ,in−1 are closed and nonempty if (i0 , . . , in−1 ) is admissible. (B). ,in−1 , j = 0, . . , p − 1. Dynamical Chaos in Terms of the -complexity 45 (C). ,in−1 → 0 as n → ∞. ,in−1 . ,in−1 ) The closed set F becomes a Cantor set, provided that the following “separation conditions” hold (D). ,jn−1 ∩F = ∅ whenever (i0 , . . , in−1 ) = (j0 , . . , jn−1 ). The coding map χ : Ω → F is deﬁned as follows: for any ω= (i0 , . . , in−1 , . ,in−1 . The simplest constructions are of Moran type.

In−1 ) The closed set F becomes a Cantor set, provided that the following “separation conditions” hold (D). ,jn−1 ∩F = ∅ whenever (i0 , . . , in−1 ) = (j0 , . . , jn−1 ). The coding map χ : Ω → F is deﬁned as follows: for any ω= (i0 , . . , in−1 , . ,in−1 . The simplest constructions are of Moran type. In this case Ω = Ωp and basic sets satisfy additional axioms. (M1). Every basic set is the closure of its interior. (M2). ,jn−1 = ∅ if (i0 , . . , in−1 ) = (j0 , . . , jn−1 ). (M3). ,in−1 .

3. Dynamical Systems with Zero Entropy This is a rich family of systems including nondispersive billiards and aperiodic transformations which display divergence of trajectories with a power-law rate and some mildly unpredictabililty in the sense of the linear prediction theory (see references in [12]) among them we shall consider mainly examples of skew products of dynamical systems. We ﬁrst deﬁne the general concept of skew products of DS. Notes on Spectral Theory, Mixing and Transport. 1 Let (X, τ, µ) be a DS, (Y, ν) be a probability measure space and for µ-almost all x ∈ X, let Sx be a ν-measure-preserving transformations on Y .