Ergodic theory with a view towards number theory
by Manfred Einsiedler and Thomas Ward
Ergodic Theory with a view towards Number Theory
Springer Graduate Text in Mathematics Volume 259
This is a project that aims to develop enough of the basic machinery of ergodic theory to describe some of the recent applications of ergodic theory to number theory. Two specific emphases are to avoid reliance on background in Lie theory and to fully prove the material needed in measure theory which goes beyond the standard texts. This will be a rigorous introduction, developing the machinery of conditional measures and expectations, mixing, and recurrence. Applications include the ergodic proof of Szemeredi's theorem and the connection between the continued fraction map and the modular surface.
Please send any comments to the authors.
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Review in Ergodic Theory and Dynamical Systems (S.G. Dani)
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Review in Jahresber Deutsch Math-Ver (Barak Weiss)
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Review in Math Reviews (Vitaly Bergelson)
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Review in Zentralblatt (Adrian Atanasiu)
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Review in Monatshefte (Viktor Losert)
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Review in European Mathematical Society (Antonio Díaz-Cano Ocaña)
A subsequent volume, Entropy in ergodic theory and topological dynamics, will continue the development. Possible future topics include a counting problem on a variety, and maybe some simple cases of the connection to integer quadratic forms in the recent work of Ellenberg and Venkatesh. Possible further chapters might address the result on Arithmetic Quantum Unique Ergodicity by Lindenstrauss.