Positive transfer operators and decay of correlations. (English) Zbl 1012.37015

Baladi, Viviane

Advanced Series in Nonlinear Dynamics. 16. Singapore: World Scientific. x, 314 p. (2000).

Publisher’s description: Although individual orbits of chaotic dynamical systems are by definition unpredictable, the average behavior of typical trajectories can often be given a precise statistical description. Indeed, there often exist ergodic invariant measures with special additional features. For a given invariant measure, and a class of observables, the correlation functions tell whether (and how fast) the system “mixes”, i.e. “forgets” its initial conditions. This book, addressed to mathematicians and mathematical (or mathematically inclined) physicists, shows how the powerful technology of transfer operators, imported from statistical physics, has been used recently to construct relevant invariant measures, and to study the speed of decay of their correlation functions, for many chaotic systems. Links with dynamical zeta functions are explained.
Contents: Subshifts of finite type: A key symbolic model; Smooth uniform expanding dynamics; Piecewise expanding systems; Hyperbolic systems. The book is intended for graduate students or researchers entering the field, and the technical prerequisites have been kept to a minimum.

MSC:

37C30	Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
37-02	Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
37D20	Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
37A25	Ergodicity, mixing, rates of mixing
47B38	Linear operators on function spaces (general)

Keywords:

statistical physics; invariant measures; speed of decay; correlation functions; chaotic systems; dynamical zeta functions; subshifts; symbolic model; expanding systems; hyperbolic systems