Ergodic Theory, Randomness and Dynamical Systems (Mathematical Monograph) - ISBN:0300017456

Ergodic Theory, Randomness and Dynamical Systems (Mathematical Monograph)

Date: 10 April 2011, 01:12

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ACKNOWLEDGMENTS
I would like to express my appreciation and gratitude to my friends, Ya. Sinai, Nathaniel A. Friedman, Meir Smorodinsky, Paul C. Shields, Roy L. Adler, Benjamin Weiss, Yitzhak Katznelson, and Giovanni Gallavotti, who
have contributed so much to the subject matter of these lectures. The isomorphism theorem for Bernoulli shifts grew out of an attempt to get a better understanding of Sinai's weak isomorphism theorem. Nathaniel Friedman
worked with me to get very substantial simplifications in the original proof of the isomorphism theorem, and we jointly got a criterion to show that certain Markov shifts were Bernoullian. This paved the way for much of the
later work. Meir Smorodinsky made further substantial simplifications in the isomorphism theorem and proved a strong lemma concerning entropy and ?-independence which enabled him to extend the isomorphism theorem
to countably infinite partitions. I have borrowed heavily from his book Entropy and Ergodic Theory. I have also borrowed heavily from Paul Shields' book The Theory of Bernoulli Shifts.* The example of a K-auto-morphism that is not Bernoulli was radically simplified—the whole idea of why it worked was changed—and this came from discussions with Roy Adler and Paul Shields. The part of these lectures on K-automorphisms is largely lifted from a joint paper with Paul Shields. The part on classical systems is completely lifted from a joint paper with Benjamin Weiss and depends on work of Sinai and Anosov. Sinai's remarkable work on the hard sphere gas gives one of the nicest applications of the theory in these lectures. Gallavotti showed me how to make this application. Yitzhak Katznelson's beautiful theorem on the automorphism of the n-dimensional torus showed what could be done by combining the theory of these lectures with more classical analytic techniques. Katznelson and Weiss extended the isomorphism theorem to the case of several commuting transformations, and Gallavotti, diLiberto, and Russo applied this to the Icing model.
This book, which gives a more detailed exposition of the underlying ideas, provides a good introduction to the subject.

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