BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY
نویسنده
چکیده
To many mathematicians, the phrase “dynamical systems” refers mainly to transformations and flows on smooth manifolds (read “smooth dynamics”). However, over time, the meaning has grown to include transformations and flows on topological spaces (read “topological dynamics”) and on measure spaces (read “ergodic theory”). Of course, in many situations, one might have more than one transformation or more than one flow. If n invertible transformations on a space commute, then one obtains an action of Z on the space. Similarly, if n invertible complete flows commute, then one has an action of R. In some situations, however, the transformations and flows may not commute, in which case one obtains an action of a more complicated group. Generally speaking, as the group under consideration attains higher levels of complexity, the possibilities for actions become fewer, since one must find transformations and flows that satisfy more and more relations. One finds that one does not always have great flexibility in choosing actions; this lack of flexibility is often called “rigidity”. From a certain perspective, terminology notwithstanding, the most complicated Lie groups are simple Lie groups. One may hope to classify how these groups may act on a variety of geometric spaces. This is R. J. Zimmer’s program. The end goal of the book under review is to give • an accessible exposition of Zimmer’s cocycle superrigidity theorem, which is an ergodic theoretic (i.e., measure theoretic) result; and • to give a new proof that works, not just in the measure theoretic category, but in the smooth category.
منابع مشابه
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY
Representations of semisimple Lie algebras in the BGG category í µí²ª, by James E.
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