# Homogeneous Flows, Moduli Spaces and Arithmetic

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*Manfred Leopold Einsiedler; David Alexandre Ellwood; Alex Eskin; Dmitry Kleinbock; Elon Lindenstrauss; Gregory Margulis; Stefano Marmi; Jean-Christophe Yoccoz*

A co-publication of the AMS and Clay Mathematics Institute

This book contains a wealth of material concerning two very active and
interconnected directions of current research at the interface of
dynamics, number theory and geometry. Examples of the dynamics
considered are the action of subgroups of
\(\mathrm{SL}(n,\mathbb{R})\) on the space of unit volume
lattices in \(\mathbb{R}^n\) and the action of
\(\mathrm{SL}(2,\mathbb{R})\) or its subgroups on moduli spaces
of flat structures with prescribed singularities on a surface of genus
\(\ge 2\).

Topics covered include the following:

(a) Unipotent flows: non-divergence, the classification of
invariant measures, equidistribution, orbit closures.

(b) Actions of higher rank diagonalizable groups and their invariant
measures, including entropy theory for such actions.

(c) Interval exchange maps and their connections to translation
surfaces, ergodicity and mixing of the Teichmüller geodesic flow,
dynamics of rational billiards.

(d) Application of homogeneous flows to arithmetic, including
applications to the distribution of values of indefinite quadratic
forms at integral points, metric Diophantine approximation,
simultaneous Diophantine approximations, counting of integral and
rational points on homogeneous varieties.

(e) Eigenfunctions of the Laplacian, entropy of quantum limits, and
arithmetic quantum unique ergodicity.

(f) Connections between equidistribution and automorphic forms and their
\(L\)-functions.

The text includes comprehensive introductions to the state-of-the-art in
these important areas and several surveys of more advanced topics,
including complete proofs of many of the fundamental theorems on the
subject. It is intended for graduate students and researchers wishing to
study these fields either for their own sake or as tools to be applied in
a variety of fields such as arithmetic, Diophantine approximations,
billiards, etc.

Titles in this series are co-published with the Clay Mathematics Institute (Cambridge, MA).

#### Readership

Graduate students and research mathematicians interested in the interface of dynamics, geometry, and number theory.