The Ergodic Theory of Lattice Subgroups (eBook)

Alexander Gorodnik, Amos Nevo (Autoren)

eBook Download: PDF

2009
160 Seiten
Princeton University Press (Verlag)
978-1-4008-3106-7 (ISBN)

Infos

The results established in this book constitute a new departure in ergodic theory and a significant expansion of its scope. Traditional ergodic theorems focused on amenable groups, and relied on the existence of an asymptotically invariant sequence in the group, the resulting maximal inequalities based on covering arguments, and the transference principle. Here, Alexander Gorodnik and Amos Nevo develop a systematic general approach to the proof of ergodic theorems for a large class of non-amenable locally compact groups and their lattice subgroups. Simple general conditions on the spectral theory of the group and the regularity of the averaging sets are formulated, which suffice to guarantee convergence to the ergodic mean. In particular, this approach gives a complete solution to the problem of establishing mean and pointwise ergodic theorems for the natural averages on semisimple algebraic groups and on their discrete lattice subgroups. Furthermore, an explicit quantitative rate of convergence to the ergodic mean is established in many cases. The topic of this volume lies at the intersection of several mathematical fields of fundamental importance. These include ergodic theory and dynamics of non-amenable groups, harmonic analysis on semisimple algebraic groups and their homogeneous spaces, quantitative non-Euclidean lattice point counting problems and their application to number theory, as well as equidistribution and non-commutative Diophantine approximation. Many examples and applications are provided in the text, demonstrating the usefulness of the results established.

Autor

Alexander Gorodnik is senior research fellow in mathematics at the University of Bristol. Amos Nevo is professor of mathematics at the Technion, Israel Institute for Technology.

Produkt-Details

• Erscheint lt. Verlag: 21.9.2009
• Reihe/Serie: Annals of Mathematics Studies
• Reihe/Serie: Annals of Mathematics Studies
• Zusatzinfo: 2 line illus.
• Verlagsort: Princeton
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
• Themenwelt: Technik
• Schlagworte: absolute continuity • algebraic group • amenable group • Asymptote • asymptotic analysis • asymptotic expansion • automorphism • Borel set • bounded function • bounded operator • Bounded set (topological vector space) • congruence subgroup • continuous function • Convergence of random variables • convolution • Coset • Counting • Counting problem (complexity) • Differentiable function • Dimension (vector space) • Diophantine approximation • Direct integral • direct product • discrete group • Embedding • Equidistribution theorem • Ergodicity • ergodic theory • estimation • Explicit formulae (L-function) • Family of sets • Haar measure • hilbert space • hyperbolic space • induced representation • Infimum and supremum • Initial Condition • interpolation theorem • Invariance principle (linguistics) • invariant measure • irreducible representation • Isometry Group • Iwasawa group • Lattice (group) • Lie algebra • Linear algebraic group • Linear space (geometry) • Lipschitz continuity • mass distribution • Mathematical Induction • Maximal compact subgroup • Maximal ergodic theorem • Measure (mathematics) • mellin transform • Metric Space • Monotonic Function • Neighbourhood (mathematics) • normal subgroup • Number Theory • One-parameter group • Operator norm • Orthogonal complement • P-adic number • parametrization • Parity (mathematics) • Pointwise • pointwise convergence • Principal homogeneous space • Principal series representation • Probability • probability measure • Probability space • Rate of Convergence • Regular representation • Representation Theory • Resolution of Singularities • Sobolev Space • Special case • Spectral Gap • spectral method • spectral theory • Square (algebra) • SUBGROUP • subsequence • Subset • symmetric space • Tensor Algebra • tensor product • Theorem • Transfer principle • Unitary Group • Unitary Representation • Unit sphere • Unit vector • Upper and lower bounds • Variable (mathematics) • Vector group • Vector Space • Volume form • word metric
• ISBN-10: 1-4008-3106-7 / 1400831067
• ISBN-13: 978-1-4008-3106-7 / 9781400831067