Chapter 9 Ergodic theory and dynamics of G-spaces (with special emphasis on rigidity phenomena)

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This chapter discusses the case of group, as opposed to semi-group, actions. There are various technical complications that arise in passing from invertible to noninvertible actions, and in general, the nonamenable noninvertible case has not been given sufficient attention as yet to warrant inclusion into a general survey. The primary focus of the survey is on those aspects of the ergodic theory and differentiable dynamics of group actions that are most distinct from the theory for R and ℤ. As ergodic theory for actions of general amenable groups share with R and ℤ many key properties, the survey discussed in the chapter is concerned in large part with the actions of nonamenable groups. Orbit equivalence provides a particularly compelling example: all finite measure-preserving ergodic actions of discrete amenable groups are orbit equivalent. For certain groups that are both “sufficiently large” and “rigid,” orbit equivalence essentially implies isomorphism.