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Dynamical Systems of Algebraic Origin (Paperback, 1995)
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Dynamical Systems of Algebraic Origin (Paperback, 1995)
Series: Modern Birkhauser Classics
Expected to ship within 10 - 15 working days
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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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