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

Errata file

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.

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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.