Titlebook: Ergodic Theory and Differentiable Dynamics; Ricardo Ma?é Book 1987 Springer-Verlag Berlin Heidelberg 1987 dynamical systems.ergodic theory

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Measure Theory,The purpose of this chapter is to present the basic definitions and theorems of measure theory. Proofs are only included when they cannot be found in standard references or when the formulation of the statement involved differs significantly from the usual one. The basic references for this chapter are Rudin [R6] and Munroe [M13].

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Measure-Preserving Maps,If (., ., .) is a measure space, we say that a measurable map .: . → . is ., and that . is . under ., if for every . ∈ . we have .(.)= .(.(.)). The dynamic behavior of measure-preserving maps is the theme of ergodic theory.

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Entropy,, ., .) and (., ., .)are equivalent. The approach developed there, involving the study of spectral properties of the associated isometric operators.: . (., ., .) → . (., ., .) (. = 1,2), led to the concept of spectral equivalence. We proved that equivalent maps are spectrally equivalent, and mention

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0071-1136 tion of whether to use the opportunity to introduce major additions. In a book like this, ending in the heart of a rich research field, there are always further topics that should arguably be included. Subjects like geodesic flows or the role of Hausdorff dimension in con- temporary ergodic theory a

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, in Biotechnological Applications, and is constant almost everywhere. Similar questions had already shown up in other areas of mathematics, for example, in the problem of the average movement of the perihelion in celestial mechanics (see Arnold [A6]).