This text is a rigorous introduction to Ergodic theory, developing the machinery of conditional measures and expectations, mixing, and recurrence. It describes some recent applications to number theory, and goes beyond the standard texts in this topic.

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.

With a rigorous development of basic ergodic theory and homogeneous dynamics, no background in Ergodic theory or Lie theory is assumed Offers both complete and motivated treatments of Weyl and Szemeredi theorems Provides a number of exercises and hints to problems Includes supplementary material: sn.pub/extras

Autorentext

Menny Aka studied at the Hebrew University, with a Ph.D. in 2012 under Alexander Lubotzky. He held research positions at EPFL and ETH Zürich before becoming a senior scientist at ETH Zürich. He works on the interaction between number theory, ergodic theory and group theory. An enthusiastic and innovative lecturer, he is interested in making mathematics accessible, especially to younger audiences. He has initiated and taught in various programs for high school students, including projects aimed at gifted students and prospective undergraduates. He is interested in showcasing the beauty and simplicity underpinning complex mathematical ideas. Manfred Einsiedler studied at the University of Vienna, with a Ph.D. in 1999 under Klaus Schmidt. He held research positions at the University of East Anglia, Penn State University, the University of Washington, and Princeton University as a Clay Research Scholar. After becoming a Professor at Ohio State University he joinedETH Zürich. In 2004 he won the Research Prize of the Austrian Mathematical Society, in 2008 he was an invited speaker at the European Mathematical Congress in Amsterdam, and in 2010 he was an invited speaker at the International Congress of Mathematicians in Hyderabad. He works on ergodic theory (especially dynamical and equidistribution problems on homogeneous spaces) and its applications to number theory. He has collaborated with Grigory Margulis and Akshay Venkatesh. With Elon Lindenstrauss and Anatole Katok, Einsiedler proved that a conjecture of Littlewood on Diophantine approximation is "almost always" true. Thomas Ward studied at the University of Warwick, with a Ph.D. in 1989 under Klaus Schmidt. He held research positions at the University of Maryland, College Park and at Ohio State University before joining the University of East Anglia in 1992. Since 2008 he has served on university executives, as Pro-Vice-Chancellor for Education at the University of East Anglia and Durham University, and since 2016 as Deputy Vice-Chancellor (Student Education) at the University of Leeds. He worked on the ergodic theory of algebraic dynamical systems, compact group automorphisms, and number theory. A long collaboration with Graham Everest on links between number theory and dynamical systems included the book "Heights of polynomials and entropy in algebraic dynamics" and a paper on Diophantine equations that won the 2012 Lester Ford Prize for mathematical exposition. With Einsiedler he has written "Ergodic theory with a view towards number theory" in 2011 and "Functional analysis, spectral theory, and applications" in 2017.

Inhalt
Motivation.- Ergodicity, Recurrence and Mixing.- Continued Fractions.- Invariant Measures for Continuous Maps.- Conditional Measures and Algebras.- Factors and Joinings.- Furstenberg's Proof of Szemeredi's Theorem.- Actions of Locally Compact Groups.- Geodesic Flow on Quotients of the Hyperbolic Plane.- Nilrotation.- More Dynamics on Quotients of the Hyperbolic Plane.- Appendix A: Measure Theory.- Appendix B: Functional Analysis.- Appendix C: Topological Groups