By Ya.G. Sinai
Following the idea that of the EMS sequence this quantity units out to familiarize the reader to the basic principles and result of smooth ergodic conception and to its functions to dynamical platforms and statistical mechanics. The exposition begins from the fundamental of the topic, introducing ergodicity, blending and entropy. Then the ergodic concept of tender dynamical platforms is gifted - hyperbolic thought, billiards, one-dimensional structures and the weather of KAM concept. various examples are offered rigorously in addition to the information underlying an important effects. The final a part of the ebook offers with the dynamical structures of statistical mechanics, and particularly with numerous kinetic equations. This publication is obligatory analyzing for all mathematicians operating during this box, or eager to know about it.
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Additional info for Dynamical systems: Ergodic theory with applications to dynamical systems and statistical mechanics
An ergodic automorphism T such that ([> is a complete system of functions in L 2 (M) is called an automorphism with quasi-discrete spectrum. Any automorphism with pure point spectrum has, obviously, quasi-discrete spectrum. The ergodic skew translation on the 2-dimensional torus: T(x, y) = (x + ex, y + x), x, Y E S1, ex is irrational, is the example of the automorphism with quasi-discrete but not pure point spectrum. e. such automorphisms T that all Tn, n # 0, are ergodic. M. Abramov [A]). Suppose T b T2 are totally ergodic automorphisms of the Lebesgue spaces (M b vII b J11)' (M2, v112,J12) with quasi-discrete spectrum.
The automorphism TE is called the induced automorphism constructed from the automorphism T and the set E. Now consider the "dual" construction. Suppose T1 is an automorphism of the space (M 1, Jt l' Jl1) and f ELl (M 1, Jt l' Jl1) is a positive integer valued function. Introduce the measure space M whose points are of the form (x l ' i), where Xl E M 1 , 0::::; i < f(xd, i is an integer. Define the measure Jl on M by Jl(A;) where Ai given by c = Jl(A)' (f M M is the set of the form (A, i), A I f dJl1) E Jt l' The transformation Tf of M -1, is an automorphism of M.
Using the Pontryagin duality theory one can easily construct for arbitrary countable subgroup A c Sl the automorphism Twith pure point spectrum such that AAT) = A. This automorphism T is a group translation on the character group M of the group A with the Haar measure Jl. The group M is compact in the case considered and T is given by Tg = g' go, (g, go E M) where go(A) = A, A E A. 1, we get the followin~ fact: any ergodic automorphism with pure point spectrum is metrically isomorphic to a certain group translation on the character group of its spectrum.
Dynamical systems: Ergodic theory with applications to dynamical systems and statistical mechanics by Ya.G. Sinai
Categories: Mathematical Physics