Dynamical Systems of Algebraic Origin

Dynamical Systems of Algebraic Origin by Klaus Schmidt.

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-...

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Bibliographic Details
Main Author: Schmidt, Klaus (Author)
Corporate Author: SpringerLink (Online service)
Format: eBook
Language:English
Published: Basel : Birkhäuser Basel : Imprint: Birkhäuser, 1995.
Edition:1st ed. 1995.
Series:Progress in Mathematics, 128
Springer eBook Collection.
Subjects:
Topological groups.
Lie groups.
Algebraic geometry.
Probabilities.
Statistical physics.
Dynamical systems.
Group theory.
Functions of real variables.
Electronic resources (E-books)
Online Access:Click to view e-book
Holy Cross Note:Loaded electronically.
Electronic access restricted to members of the Holy Cross Community.
Table of Contents:
  • I. Group actions by automorphisms of compact groups
  • 1. Ergodicity and mixing
  • 2. Expansiveness and Lie subshifts
  • 3. The descending chain condition
  • 4. Groups of Markov type
  • II. ?d-actions on compact abelian groups
  • 5. The dual module
  • 6. The dynamical system defined by a Noetherian module
  • 7. The dynamical system defined by a point
  • 8. The dynamical system defined by a prime ideal
  • III. Expansive automorphisms of compact groups
  • 9. Expansive automorphisms of compact connected groups
  • 10. The structure of expansive automorphisms
  • IV. Periodic points
  • 11. Periodic points of ?d-actions
  • 12. Periodic points of ergodic group automorphisms
  • V. Entropy
  • 13. Entropy of ?d-actions
  • 14. Yuzvinskii’s addition formula
  • 15. ?d-actions on groups with zero-dimensional centres
  • 16. Mahler measure
  • 17. Mahler measure and entropy of group automorphisms
  • 18. Mahler measure and entropy of ?d-actions
  • VI. Positive entropy
  • 19. Positive entropy
  • 20. Completely positive entropy
  • 21. Entropy and periodic points
  • 22. The distribution of periodic points
  • 23. Bernoullicity
  • VII. Zero entropy
  • 24. Entropy and dimension
  • 25. Shift-invariant subgroups of $$ {(\mathbb{Z}/p\mathbb{Z})̂{{{\mathbb{Z}̂2}}}} $$
  • 26. Relative entropies and residual sigma-algebras
  • VIII. Mixing
  • 27. Multiple mixing and additive relations in fields
  • 28. Masser’s theorem and non-mixing sets
  • IX. Rigidity
  • 29. Almost minimal ?d-actions and invariant measures
  • 30. Cohomological rigidity
  • 31. Isomorphism rigidity.

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