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Evolution Semigroups in Dynamical Systems and Differential Equations
Carmen Chicone and Yuri Latushkin, University of Missouri, Columbia, MO
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Mathematical Surveys and Monographs
1999; 361 pp; hardcover
Volume: 70
ISBN-10: 0-8218-1185-1
ISBN-13: 978-0-8218-1185-6
List Price: US$91
Member Price: US$72.80
Order Code: SURV/70
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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.

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

Table of Contents

  • Introduction
  • Semigroups on Banach spaces and evolution semigroups
  • Evolution families and Howland semigroups
  • Characterizations of dichotomy for evolution families
  • Two applications of evolution semigroups
  • Linear skew-product flows and Mather evolution semigroups
  • Characterizations of dichotomy for linear skew-product flows
  • Evolution operators and exact Lyapunov exponents
  • Bibliography
  • List of notations
  • Index
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