Dynamical Systems of Algebraic Origin (Progress in Mathematics) - Hardcover

Schmidt, Klaus

Hardcover

ISBN 10: 3764351748 ISBN 13: 9783764351748

Publisher: Birkhäuser, 1995

Synopsis

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.

About this title

Synopsis

Although much of classical ergodic theory is concerned with single transformations and one-paramter flows, the subject inherits from statistical mechanics not only its name, but also an obligation to analyze spatially extended systems with multi-dimensional symmetry groups. However, the wealth of concrete and natural examples which have contributed to the appeal and development of classical dynamics,is noticeably absent in this more general theory. This book aims to remedy this scarcity by introducing a class of continuous diverse ZZd-actions diverse enough to exhibit many of the new phenomena encountered in the transition from ZZ to ZZd, but which nevertheless lends itself to systematic study: the ZZd-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 ZZd-actions.

Edition details

• Publisher: Birkhäuser
• Publication date: 1995
• Language: English
• ISBN 10: 3764351748
• ISBN 13: 9783764351748
• Binding: Hardcover
• Edition number: 1
• Number of pages: 310