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This volume presents selected papers by the brilliant Uruguayan
mathematician Ricardo Mane, known for his outstanding contributions
to dynamical systems and ergodic theory. He was a student of Jacob
Palis at IMPA and pursued his whole scientific career at IMPA. Mane
was invited to speak twice in the section of Ordinary Differential
Equations and Dynamical Systems, at the International Congress of
Mathematics held in Warsaw in 1983 and in Zurich in 1994. He was
also a speaker at the Colloquium organized by the Societe
Mathematique de France, celebrating R. Thom's 65 anniversary. In
1994, he became a member of the Brazilian Academy of Sciences and
was awarded the Third World Academy of Sciences Prize for
Mathematics.
This version differs from the Portuguese edition only in a few
additions and many minor corrections. Naturally, this edition
raised the question 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 are two of the
most tempting gaps to fill. However, I let it stand with
practically the same boundaries as the original version, still
believing these adequately fulfill its goal of presenting the basic
knowledge required to approach the research area of Differentiable
Ergodic Theory. I wish to thank Dr. Levy for the excellent
translation and several of the correc tions mentioned above. Rio de
Janeiro, January 1987 Ricardo Mane Introduction This book is an
introduction to ergodic theory, with emphasis on its relationship
with the theory of differentiable dynamical systems, which is
sometimes called differentiable ergodic theory. Chapter 0, a quick
review of measure theory, is included as a reference. Proofs are
omitted, except for some results on derivatives with respect to
sequences of partitions, which are not generally found in standard
texts on measure and integration theory and tend to be lost within
a much wider framework in more advanced texts."
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