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Typical dynamics of volume preserving homeomorphisms. (English) Zbl 0970.37001
Cambridge Tracts in Mathematics. 139. Cambridge: Cambridge University Press. xix, 216 p. (2000).
The two authors of this monograph have, over the last twenty five years, contributed a vast amount generalizing the classical results of John Oxtoby and Stan Ulam on the typical behaviour of manifold homeomorphisms which preserve a fixed measure. This monograph puts these results, and others, into a logical framework.
More precisely, the book starts by looking at the case where the compact manifold is simply the unit \(n\)-dimensional cube, endowed with Lebesgue measure. Then they show how the results for the cube can be extended to arbitrary compact manifolds. The authors then go on to establish which results can be extended to non-compact manifolds where the measure may be infinite.
The book is comprehensively written in the sense that very little prior knowledge of ergodic theory or measure theory is required to follow the text.

37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
37A05 Dynamical aspects of measure-preserving transformations