Dynamical Systems of Algebraic Origin / Edition 1

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Overview

Although the study of dynamical systems is mainly concerned with single trans­ formations and one-parameter flows (i. e. with actions of Z, N, JR, or JR+), er­ godic theory inherits from statistical mechanics not only its name, but also an obligation to analyze spatially extended systems with multi-dimensional sym­ metry groups. However, the wealth of concrete and natural examples, which has contributed so much to the appeal and development of classical dynamics, is noticeably absent in this more general theory. A remarkable exception is provided by a class of geometric actions of (discrete subgroups of) semi-simple Lie groups, which have led to the discovery of one of the most striking new phenomena in multi-dimensional ergodic theory: under suitable circumstances orbit equivalence of such actions implies not only measurable conjugacy, but the conjugating map itself has to be extremely well behaved. Some of these rigidity properties are inherited by certain abelian subgroups of these groups, but the very special nature of the actions involved does not allow any general conjectures about actions of multi-dimensional abelian groups. Beyond commuting group rotations, commuting toral automorphisms and certain other algebraic examples (cf. [39]) it is quite difficult to find non-trivial smooth Zd-actions on finite-dimensional manifolds. In addition to scarcity, these examples give rise to actions with zero entropy, since smooth Zd-actions with positive entropy cannot exist on finite-dimensional, connected manifolds. Cellular automata (i. e.

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Editorial Reviews

Booknews
Remedies the scarcity of explicit examples in general areas of ergodic theory by introducing a class of continuous Zd-actions diverse enough to exhibit many of the new phenomena encountered in the transition from Z to Zd, but which nevertheless lends itself to systematic study: the Zd-actions by automorphisms of compact, abelian groups. Topics include group actions by automorphisms of compact groups; Zd-actions on compact abelian groups; expansive automorphisms of compact groups; periodic points; entropy; positive entropy; zero entropy; mixing; and rigidity. Annotation c. Book News, Inc., Portland, OR (booknews.com)
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Product Details

  • ISBN-13: 9783034802765
  • Publisher: Springer Basel
  • Publication date: 1/5/2012
  • Series: Modern Birkhauser Classics Series
  • Edition description: 1995
  • Edition number: 1
  • Pages: 310
  • Product dimensions: 6.14 (w) x 9.21 (h) x 0.69 (d)

Meet the Author

Klaus Schmidt is a Professor of Mathematics at the University of Vienna, Austria.

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Table of Contents

Introduction
Ch. I Group actions by automorphisms of compact groups
Ch. II Z[superscript d]-actions on compact abelian groups
Ch. III Expansive automorphisms of compact groups
Ch. IV Periodic points
Ch. V Entropy
Ch. VI Positive entropy
Ch. VII Zero entropy
Ch. VIII Mixing
Ch. IX Rigidity
Bibliography
Index
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