##
**Dynamical systems and processes.**
*(English)*
Zbl 1179.37002

IRMA Lectures in Mathematics and Theoretical Physics 14. Zürich: European Mathematical Society (ISBN 978-3-03719-046-3/hbk). xii, 761 p. (2009).

This book displays mathematics written for physicists: the topic of dynamical systems is viewed mainly from the point of view of ergodicity. Various tools and methods arising from spectral theory, ergodic theory and probability theory are presented within a common setting and are applied to almost everywhere convergence problems.

Consequently the book is organized in 4 parts: I. Spectral theorems and convergence in mean. II. Ergodic theorems. III. Methods arising from the theory of stochastic processes. IV. Three studies (dedicated to applications).

The problems arising from spectral theory deal with the von Neumann theorem, spectral regularization and spectral representation of weakly stationary processes. The topics of ergodicity are: dynamical systems – ergodicity and mixing; pointwise ergodic theorems; Banach and continuity principle; maximal operators ans Gaussian processes; the central limit theorem for dynamical systems. From the theory of stochastic processes there are considered: the metric entropy method, the majorizing method; Gaussian processes. The three studies are concerned with Riemann sums and two other mathematical problems. It appears that the book is more mathematics than physics oriented but we still believe it is useful for both classes of experts.

Consequently the book is organized in 4 parts: I. Spectral theorems and convergence in mean. II. Ergodic theorems. III. Methods arising from the theory of stochastic processes. IV. Three studies (dedicated to applications).

The problems arising from spectral theory deal with the von Neumann theorem, spectral regularization and spectral representation of weakly stationary processes. The topics of ergodicity are: dynamical systems – ergodicity and mixing; pointwise ergodic theorems; Banach and continuity principle; maximal operators ans Gaussian processes; the central limit theorem for dynamical systems. From the theory of stochastic processes there are considered: the metric entropy method, the majorizing method; Gaussian processes. The three studies are concerned with Riemann sums and two other mathematical problems. It appears that the book is more mathematics than physics oriented but we still believe it is useful for both classes of experts.

Reviewer: Vladimir Răsvan (Craiova)

### MSC:

37-02 | Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory |

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |