# Project

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## Equidistribution and dynamics on homogeneous spaces

 Applicant Einsiedler Manfred 152819 Project funding (Div. I-III) Departement Mathematik ETH Zürich ETH Zurich - ETHZ Mathematics 01.05.2014 - 30.04.2018 770'000.00
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### Keywords (6)

Equidistribution; Homogeneous spaces; Dynamical systems; Number theory; Effectiveness; Cartan action

### Lay Summary (German)

Das mathematische Gebiet der dynamischen Systeme ist aus der Untersuchung von dem Langzeitverhalten von physikalischen Systemen (z.B. Bewegungen der Planeten) entstanden. Ein weiteres Gebiet mit fundamentaler Bedeutung für die Mathematik, die Physik, und auch anderen Anwendungen, ist die Theorie der Lie Gruppen, welche alle möglichen Geometrien beschreibt. In der Untersuchung dynamischer Systeme auf homogenen Räumen werden diese beiden Theorien vereint in dem man dynamische Systeme in allgemeineren Geometrien untersucht. Dieses Gebiet hat starke Zusammenhänge zur Zahlentheorie, da auch in der Zahlentheorie solche allgemeineren Geometrien eine grosse Rolle spielen. Die bekannte Theorie der dynamischen Systeme auf homogenen Räumen enthält überraschend starke Aussagen über das Langzeitverhalten der Bewegungen. Die meisten dieser Resultate sind allerdings nicht effektiv, d.h. diese machen keine Aussage wie lange man warten muss bis ein gewisser Ort durch die Bewegung erreicht wird.
Lay summary

Inhalt und Ziel des Forschungsprojekts

Ein grosser Teil des Forschungsprojekts beschäftigt sich mit der Frage der Effektivität in der Theorie gewisser dynamische Systeme (definiert durch sogenannte halb-einfache Untergruppen) auf homogenen Räumen. Die Zahlentheorie stellt hier ein wichtiges Hilfsmittel zur Verfügung, denn vor ca. 10 Jahren wurde gezeigt, dass die zahlentheoretisch interessanten homogenen Räume sehr starke Zusammenhangseigenschaften haben. Diese machen es einem möglich effektive Beweise durchzuführen.  Viele der möglichen Anwendungen solcher effektiven Sätze werden in der Zahlentheorie liegen, aber weitere Anwendungen in Richtung der Physik sind auch zu erwarten.

Es gibt aber noch weitere dynamische Systeme auf homogenen Räumen (definiert durch sogenannte diagonalisierbare Untergruppen) für die keine so starken Aussagen über das Langzeitverhalten von Bahnen bekannt sind (oder zum Teil auch nicht zu erwarten sind). Für diese dynamischen Systeme sollen stärkere Sätze bewiesen werden, welche wiederum Anwendungen in der Zahlentheorie und Physik haben sollte.

 Direct link to Lay Summary Last update: 15.04.2014

### Responsible applicant and co-applicants

Name Institute
 Einsiedler Manfred Departement Mathematik ETH Zürich

### Employees

Name Institute
 Wieser Andreas Departement Mathematik ETH Zürich
 Rühr Rene Tel Aviv University School of Mathematical Sciences
 Sert Cagri Institut für Mathematik Universität Zürich

### Publications

Publication
Aka Menny, Einsiedler Manfred, Li Han, Mohammadi Amir (2020), On effective equidistribution for quotients of SL(d,ℝ), in Israel Journal of Mathematics, 236(1), 365-391.
EINSIEDLER MANFRED, MAIER ALEX (2020), Simultaneous equidistributing and non-dense points for non-commuting toral automorphisms, in Ergodic Theory and Dynamical Systems, 40(1), 175-193.
Einsiedler Manfred, Lindenstrauss Elon, Mohammadi Amir (2020), Diagonal actions in positive characteristic, in Duke Mathematical Journal, 169(1), 117-175.
Einsiedler M., Margulis G., Mohammadi A., Venkatesh A. (2020), Effective equidistribution and property $(\tau )$, in Journal of the American Mathematical Society, 33(1), 223-289.
Einsiedler Manfred, Lindenstrauss Elon (2019), Joinings of higher rank torus actions on homogeneous spaces, in Publications mathématiques de l'IHÉS, 129(1), 83-127.
EINSIEDLER MANFRED, RÜHR RENÉ, WIRTH PHILIPP (2019), Distribution of shapes of orthogonal lattices, in Ergodic Theory and Dynamical Systems, 39(06), 1531-1607.
Lytle Beverly, Maier Alex (2018), Simultaneous dense and nondense orbits for noncommuting toral endomorphisms, in Monatsh. Math., 185(3), 473-488.
Einsiedler Manfred, Lindenstrauss Elon (2018), Symmetry of entropy in higher rank diagonalizable actions and measure classification, in Journal of Modern Dynamics, 13(1), 163-185.
Einsiedler Manfred, Ghosh Anish, Lytle Beverly (2016), Badly approximable vectors, C^1 curves and number fields, in Ergodic Theory Dynam. Systems, 36(6), 1851-1864.
Einsiedler Manfred, Mozes Shahar (2016), Divisibility properties of higher rank lattices, in Transform. Groups, 21(4), 1039-1062.
Aka Menny, Einsiedler Manfred (2016), Duke's theorem for subcollections, in Ergodic Theory Dynam. Systems, 36(2), 335-342.
Rühr Rene (2016), Effectivity of uniqueness of the maximal entropy measure on p-adic homogeneous spaces, in Ergodic Theory Dynam. Systems, 36(6), 1972-1988.
Einsiedler Manfred, Mozes Shahar, Shah Nimish, Shapira Uri (2016), Equidistribution of primitive rational points on expanding horospheres, in Compos. Math., 152(4), 667-692.
Aka Menny, Einsiedler Manfred, Shapira Uri (2016), Integer points on spheres and their orthogonal grids, in J. Lond. Math. Soc. (2), 93(1), 143-158.
Aka Menny, Einsiedler Manfred, Shapira Uri (2016), Integer points on spheres and their orthogonal lattices, in Invent. Math., 206(2), 379-396.
Einsiedler M., Kadyrov S., Pohl A. (2015), Escape of mass and entropy for diagonal flows in real rank one situations, in Israel J. Math., 210(1), 245-295.
Einsiedler Manfred, Lindenstrauss Elon (2015), On measures invariant under tori on quotients of semisimple groups, in Ann. of Math. (2), 181(3), 993-1031.
Badziahin Dmitry, Bugeaud Yann, Einsiedler Manfred, Kleinbock Dmitry (2015), On the complexity of a putative counterexample to the p-adic Littlewood conjecture, in Compos. Math., 151(9), 1647-1662.
Bergelson Vitaly, Einsiedler Manfred, Tseng Jimmy (2015), Simultaneous dense and nondense orbits for commuting maps, in Israel J. Math., 210(1), 23-45.

### Collaboration

Group / person Country
Types of collaboration
 A. Mohammadi United States of America (North America)
 - in-depth/constructive exchanges on approaches, methods or results- Publication
 A. Venkatesh United States of America (North America)
 - Publication
 G. Margulis United States of America (North America)
 - Publication
 E. Lindenstrauss Israel (Asia)
 - in-depth/constructive exchanges on approaches, methods or results- Publication

### Scientific events

#### Active participation

Title Type of contribution Title of article or contribution Date Place Persons involved
 New Methods for Zimmer's Conjecture Talk given at a conference Measure Rigidity for diagonalisable actions 22.02.2018 Los Angeles, United States of America Einsiedler Manfred;
 5th Miniworkshop on Operator Theoretic Aspects of Ergodic Theory Talk given at a conference On equidistribution of two-dimensional planes / primitive rational points 17.11.2017 Tübingen, Germany Luethi Manuel; Wieser Andreas;
 Workshop on Diophantine approximation and related fields: York 2017 Talk given at a conference Measure Rigidity of diagonal actions and Diophantine approximation 26.06.2017 York, Great Britain and Northern Ireland Einsiedler Manfred;
 4th miniworkshop on Operator Theoretic Aspects of Ergodic Theory Talk given at a conference Joinings and orthogonal lattices 05.05.2017 Feldkirch, Austria Einsiedler Manfred;
 Ergodic Theory and its Connections with Arithmetic and Combinatorics Talk given at a conference Measure Rigidy and Quantitative Recurrence 12.12.2016 Marseille, Luminy, France Einsiedler Manfred;
 Dynamics, Geometry and Number Theory Talk given at a conference Measure rigidity for diagonal actions on quotients of SL(3,R)xSL(3,R) 13.06.2016 Paris, France Einsiedler Manfred;
 Renormalization in Dynamics - Pisa 2016 Talk given at a conference Measure rigidity for diagonalisable flows 04.04.2016 Pisa, Italy Einsiedler Manfred;
 Groups, Orbits and Diophantine Approximation Talk given at a conference Transportation of Positive Entropy 01.02.2016 Goa, India Einsiedler Manfred;
 Ergodic Theory, Fractals and Groups Talk given at a conference Positive entropy and (multiple) quantitative unipotent recurrence 11.10.2015 Jerusalem, Israel Einsiedler Manfred;
 NUMBER THEORY AND DYNAMICS Talk given at a conference Arithmetic Quantum Unique Ergodicity 06.07.2015 Djursholm, Sweden Einsiedler Manfred;
 Ergodic Theory and Combinatorics Conference Talk given at a conference Equidistribution of integer points on spheres and their orthogonal lattices 08.06.2015 Kristiansand, Norway Einsiedler Manfred;
 Introductory Workshop: Geometric and Arithmetic Aspects of Homogeneous Dynamics Talk given at a conference Rigidity of higher rank diagonalisable actions 02.02.2015 Berkeley, United States of America Einsiedler Manfred;
 Yu.V.Linnik Centennial Conference Talk given at a conference Joint equidistribution of primitive integer points on spheres and the shape of their orthogonal complement. 15.09.2014 St. Petersburg, Russia Einsiedler Manfred;
 Recent Progress in Dynamical Systems and Related Topics Talk given at a conference Joinings of higher rank actions, Integer points on spheres and their orthogonal complement 11.08.2014 Banff, Canada Einsiedler Manfred;

#### Self-organised

Title Date Place
 Ergodic Theory: Numbers, Fractals, and Geometry 24.09.2017 Oxford, Great Britain and Northern Ireland

### Associated projects

Number Title Start Funding scheme
 127145 Effective equidistribution on homogeneous spaces 01.10.2009 Project funding (Div. I-III)
 178958 Dynamics on homogeneous spaces and number theory 01.05.2018 Project funding (Div. I-III)

### Abstract

Equidistribution problems on homogeneous spaces are a central part of research in mathematics. One reason for the significance of these problems comes from the fact that they are intimately linked to different areas of mathematics. For instance the dynamical study of equidistribution problems on homogeneous spaces has become a powerful technique to address a variety of number theoretic problems; especially those involving Diophantine solutions to equations and inequalities - where the solution set and their equations both admit large symmetry groups.A very interesting equidistribution problem that has attracted a lot of attention since the work of Linnik from around 1960 is the distribution of integer points on large spheres with the origin at the center. Linnik established the equidistribution of the directions of these points unconditionally for d = 4 dimensions and under some mild congruence conditions for d = 3. Duke finally proved the case d = 3 unconditionally using subconvexity results on L-functions in 1988. A refinement of these results is to study the direction of the vector simultaneously with the shape of the lattice in the orthogonal complement. Once more the case of d = 3 is hardest, but as we explain in this proposal dynamics on homogeneous spaces can be used to analyze this problem for all dimensions d = 4 and also partially for d = 3.In the case of d = 4 the result by Mozes and Shah on the equidistribution of homogeneous measures is related to this problem. The Mozes-Shah result is a corollary of the theorems by Ratner on the dynamics of unipotent flows from around 1990. The theorems of Ratner, Mozes and Shah, Dani and Margulis in this area have found numerous applications to number theory and dynamics. Often this method leads to theorems that are unavailable by other means, but the disadvantage is that these results are non-effective. In a joint work the applicant, Margulis, and Venkatesh have obtained an effective equidistribution theorem by combining dynamical, number theoretical, and spectral methods. The theorem is currently restricted to the case of a real homogeneous space, and more importantly still has a technical (presumably avoidable) assumption that precludes some applications.More recently a second class of theorems have become useful for applications to number theory. These concern the dynamics of higher rank diagonalizable flows, and it is conjectured by Furstenberg, Margulis and others that these flows have similar rigidity properties as unipotent flows. The applicant has obtained in joint work with A. Katok and with E. Lindenstrauss a partial classification of positive entropy measures for higher rank diagonalizable flows. The assumption of positive entropy is crucial for the currently available techniques but even with this assumption there are still cases that have not been resolved and would have number theoretical applications.The main aim of this project is to extend the joint work of the applicant, Margulis, and Venkatesh as well as the work with E. Lindenstrauss and to apply these to number theoretic problems.
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