Ergodic Theory with a view towards Number Theory

Tom Ward

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Ergodic Theory with a view towards Number Theory

Manfred Einsiedler and Thomas Ward

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.

This web page contains the chapters in draft form in pdf format, and a copy of the entire text. All the pdf files will be removed during February 2010, and the book will appear as a Springer Graduate Text in Mathematics during 2010.

Please send any comments to the authors (if possible mentioning the date of the file you are using), or .

Entire text (viewable format with links, January 2010; 3.6MB)
Introduction
Chapter 1: Motivation (January 2010)
Chapter 2: Ergodicity, Recurrence and Mixing (January 2010)
Chapter 3: Continued Fractions (January 2010)
Chapter 4: Invariant Measures for Continuous Maps (January 2010)
Chapter 5: Conditional Measures and Algebras (January 2010)
Chapter 6: Factors and Joinings (January 2010)
Chapter 7: Furstenberg's Proof of Szemeredi's theorem (January 2010)
Chapter 8: Actions of Locally Compact Groups (January 2010)
Chapter 9: Geodesic Flow on Quotients of the Hyperbolic Plane (January 2010)
Chapter 10: Nilrotation (January 2010)
Chapter 11: More Dynamics on quotients of the hyperbolic plane (January 2010)
Appendices, References and Index (January 2010)

A subsequent volume, Entropy in ergodic theory and homogeneous 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.

University of East Anglia Norwich NR4 7TJ UK
Telephone: (+44) (0) 1603 456161 Fax: (+44) (0) 1603 458553

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Ergodic Theory with a view towards Number Theory

Manfred Einsiedler and Thomas Ward