**Mathematical Surveys and Monographs**

Volume: 70;
1999;
361 pp;
Hardcover

MSC: Primary 47; 34;
Secondary 58

Print ISBN: 978-0-8218-1185-6

Product Code: SURV/70

List Price: $96.00

Individual Member Price: $76.80

**Electronic ISBN: 978-1-4704-1297-5
Product Code: SURV/70.E**

List Price: $96.00

Individual Member Price: $76.80

# Evolution Semigroups in Dynamical Systems and Differential Equations

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*Carmen Chicone; Yuri Latushkin*

The main theme of the book is the spectral theory for
evolution operators and evolution semigroups, a subject tracing its
origins to the classical results of J. Mather on hyperbolic dynamical
systems and J. Howland on nonautonomous Cauchy problems. The authors
use a wide range of methods and offer a unique presentation.

The authors give a unifying approach for a study of
infinite-dimensional nonautonomous problems, which is based on the
consistent use of evolution semigroups. This unifying idea connects
various questions in stability of semigroups, infinite-dimensional
hyperbolic linear skew-product flows, translation Banach algebras,
transfer operators, stability radii in control theory, Lyapunov
exponents, magneto-dynamics and hydro-dynamics. Thus the book is much
broader in scope than existing books on asymptotic behavior of
semigroups.

Included is a solid collection of examples from different areas of
analysis, PDEs, and dynamical systems. This is the first monograph
where the spectral theory of infinite dimensional linear skew-product
flows is described together with its connection to the multiplicative
ergodic theorem; the same technique is used to study evolution
semigroups, kinematic dynamos, and Ruelle operators; the theory of
stability radii, an important concept in control theory, is also
presented. Examples are included and non-traditional applications are
provided.

#### Table of Contents

# Table of Contents

## Evolution Semigroups in Dynamical Systems and Differential Equations

- Contents ix10 free
- Chapter 1. Introduction 112 free
- Chapter 2. Semigroups on Banach Spaces and Evolution Semigroups 2132
- Chapter 3. Evolution Families and Howland Semigroups 5768
- Chapter 4. Characterizations of Dichotomy for Evolution Families 8798
- Chapter 5. Two Applications of Evolution Semigroups 131142
- Chapter 6. Linear Skew-Product Flows and Mather Evolution Semigroups 163174
- Chapter 7. Characterizations of Dichotomy for Linear Skew-Product Flows 205216
- Chapter 8. Evolution Operators and Exact Lyapunov Exponents 261272
- Bibliography 341352
- List of Notations 357368
- Index 359370

#### Readership

Graduate students and research mathematicians interested in the theory of strongly continuous semigroups of linear operators and evolution equations, Banach and \(C^*\)-algebras, infinite-dimensional and hyperbolic dynamical systems, control theory and ergodic theory; engineers, and physicists interested in Lyapunov exponents, transfer operators, etc.

#### Reviews

It was a pleasure to read this monograph, which is written in an agreeable and consistent style … This excellent exposition should serve as a reference book for further research in these fields employing the powerful methods presented by Chicone and Latushkin.

-- Mathematical Reviews