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Topological Dynamics and Applications
Edited by: M. G. Nerurkar, Rutgers University, Camden, NJ, D. P. Dokken, University of St. Thomas, St. Paul, MN, and D. B. Ellis, Beloit College, WI
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Contemporary Mathematics
1998; 334 pp; softcover
Volume: 215
ISBN-10: 0-8218-0608-4
ISBN-13: 978-0-8218-0608-1
List Price: US$84
Member Price: US$67.20
Order Code: CONM/215
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This book is a very readable exposition of the modern theory of topological dynamics and presents diverse applications to such areas as ergodic theory, combinatorial number theory and differential equations. There are three parts: 1) The abstract theory of topological dynamics is discussed, including a comprehensive survey by Furstenberg and Glasner on the work and influence of R. Ellis. Presented in book form for the first time are new topics in the theory of dynamical systems, such as weak almost-periodicity, hidden eigenvalues, a natural family of factors and topological analogues of ergodic decomposition. 2) The power of abstract techniques is demonstrated by giving a very wide range of applications to areas of ergodic theory, combinatorial number theory, random walks on groups and others. 3) Applications to non-autonomous linear differential equations are shown. Exposition on recent results about Floquet theory, bifurcation theory and Lyapanov exponents is given.

Readership

Graduate students and research mathematicians working in ergodic theory, topological dynamics, differential equations and dynamical systems.

Table of Contents

Part I. Topological Dynamics: Abstract Theory
  • H. Furstenberg and E. Glasner -- Robert Ellis and the algebra of dynamical systems
  • J. Auslander -- Weak mixing and pure weak mixing minimal flows
  • E. Glasner, M. Mentzen, and A. Siemaszko -- A natural family of factors for minimal flows
  • E. Akin and E. Glasner -- Topological ergodic decomposition and homogeneous flows
  • D. Penazzi -- On the proximal and regionally proximal relation of an extension between minimal flows
  • E. Akin, J. Auslander, and K. Berg -- Almost equicontinuity and the enveloping semigroup
  • V. Pestov -- Some universal constructions in abstract topological dynamics
  • T. Downarowicz -- Weakly almost periodic flows and hidden eigenvalues
  • E. Akin -- Enveloping linear maps
  • H. Keynes, K. Madden, N. Markley, and M. Sears -- An overview of the construction of suspension flows using continuous cocycles
  • D. Ellis -- Suspensions, inheritance, and flows on homogeneous spaces
  • H. Ikeshoji and T. Wu -- On the lifting of transformation semigroups
Part II. Applications and Other Dynamical Results
  • D. Dokken -- Idempotent measures associated to a locally compact topological group
  • S. Adams -- Another proof of Moore's ergodicity theorem for \(SL(2,\mathbb{R})\)
  • B. Weiss -- Multiple recurrence and doubly minimal systems
  • H. Furstenberg and E. Glasner -- Subset dynamics and van der Waerden's theorem
  • V. Bergelson and R. McCutcheon -- Recurrence for semigroup actions and a non-commutative Schur theorem
  • Z. Coelho, W. Parry, and R. Williams -- A note on Livšic's periodic point theorem
  • E. Glasner and J. L. King -- A zero-one law for dynamical properties
  • D. Rudolph -- Residuality and orbit equivalence
  • J. Feldman -- Uncountably many Vershik-inequivalent group actions of equal entropy
Part III. Applications to Differential Equations
  • M. Nerurkar -- Positive exponents for a dense set of continuous \(SL(2, {\mathbf R})\) valued cocycles which arise as solutions to strongly accessible linear differential systems
  • G. Sell, W. Shen, and Y. Yi -- Topological dynamics and differential equations
  • S. Novo and R. Obaya -- An ergodic and topological approach to almost periodic bidimensional linear systems
  • R. Johnson -- An application of topological dynamics to bifurcation theory
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