Ergodic Theory: with a view towards Number Theory

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Springer Science & Business Media, Sep 11, 2010 - Mathematics - 481 pages

This text is a rigorous introduction to ergodic theory, developing the machinery of conditional measures and expectations, mixing, and recurrence.

Beginning by developing the basics of ergodic theory and progressing to describe some recent applications to number theory, this book goes beyond the standard texts in this topic. Applications include Weyl's polynomial equidistribution theorem, the ergodic proof of Szemeredi's theorem, the connection between the continued fraction map and the modular surface, and a proof of the equidistribution of horocycle orbits.

Ergodic Theory with a view towards Number Theory will appeal to mathematicians with some standard background in measure theory and functional analysis. No background in ergodic theory or Lie theory is assumed, and a number of exercises and hints to problems are included, making this the perfect companion for graduate students and researchers in ergodic theory, homogenous dynamics or number theory.

 

Contents

Motivation
1
Ergodicity Recurrence and Mixing
13
Continued Fractions
69
Invariant Measures for Continuous Maps
96
Conditional Measures and Algebras
121
Factors and Joinings
152
Furstenbergs Proof of Szemerédis Theorem
171
Actions of Locally Compact Groups
231
More Dynamics on Quotients of the Hyperbolic Plane
347
Measure Theory
403
Functional Analysis
416
Topological Groups
429
Hints for Selected Exercises
441
References
447
Author Index
463
Index of Notation
467

Geodesic Flow on Quotients of the Hyperbolic Plane
276
Nilrotation
331

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