Dynamical Systems of Algebraic Origin / Edition 1

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Overview

Although much of classical ergodic theory is concerned with single transformations and one-parameter flows, the subject inherits from statistical mechanics not only its name, but also an obligation to analyze spatially extended systems with multidimensional symmetry 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. The purpose of this book is to help remedy this scarcity of explicit examples 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. One aspect of these actions, not surprising in itself but quite striking in its extent and depth nonetheless, is the connection with commutative algebra and arithmetical algebraic geometry. The algebraic framework resulting from this connection allows the construction of examples with a variety of specified dynamical properties, and by combining algebraic and dynamical tools one obtains a quite detailed understanding of this class of Zd-actions.

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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: 1/2/2012
  • 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.- Chapter I. Group actions by automorphisms fo compact groups.- Chapter II. Zd-actions on compact abelian groups.- Chapter III. Expansive automorphisms of compact groups.- Chapter IV. Periodic points.- Chapter V. Entropy.- Chapter VI. Positive entropy.- Chapter VII. Zero entropy.- Chapter VIII. Mixing.- Chapter IX. Rigidity.- Bibliography.- Index.

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