Invariant measures and arithmetic quantum unique ergodicity

By Elon Lindenstrauss. 2006

447 citations recorded by the source corpus.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/dc745047818f2db79bb719b407d4e8deb3fcd22d

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Rigidity of measures—The high entropy case and non-commuting foliations (2005, methodology): “…a left action on X = Gs/F by letting an(gF) = (a(n)g)F where gF E X. Vol. 148, 2005 RIGIDITY OF MEASURES 177 Furthermore, a gives rise to an Zk-action 0 by automorphisms of Gs and an adjoint Zk-action A on gs by letting On(g) = anga -n and A n = Ads(n) for g C Gs and n E Z k, see Section 3.2.”
Quantum unique ergodicity for SL_2(Z)\H (2009, methodology): “As remarked in the introduction, Lindenstrauss has shown that any weak-∗ limit of the micro-local lifts of Hecke-Maass forms is a constant c (in [0, 1]) times the normalized volume measure on Y .”
Invariant measures and the set of exceptions to Littlewood's conjecture (2006, methodology): “In this case we employ the techniques developed by the third named author in [20].”
DISTRIBUTION MODULO ONE AND RATNER’S THEOREM (2006, background): “Examples are proofs of quantitative versions of the Oppenheim conjecture [3], related questions on the spacings between the values of quadratic forms [4, 8, 9], a proof of quantum unique ergodicity for certain classes of hyperbolic surfaces [6], and an approach to the Littlewood conjecture on the no”
ENTROPY AND THE LOCALIZATION OF EIGENFUNCTIONS. (2008, background): “Remember that one of the ingredients of Elon Lindenstrauss’ work [Li03] in the arithmetic situation was an estimate on the entropy of semiclassical measures, proven previously by Bourgain and Lindenstrauss [BLi03].”
The distribution of periodic torus orbits on homogeneous spaces (2006, methodology): “More recently, two methods were developed [9, 10], [24] for the homogeneous case and together they were sufficient to show that for the R-split Cartan action on SL(n,Z)\ SL(n,R) positive entropy for some element is sufficient to deduce that the measure is the volume measure of the homogeneous space ”
Measure rigidity and $p$-adic Littlewood-type problems (2005, methodology): “The complete answer for positive entropy measures was given in [EKL06, Theorem 1.3] where the two separately developed methods from [EK03] and [Lin06] were combined.”
The Eigenvector Moment Flow and Local Quantum Unique Ergodicity (2013, background): “One case for which QUE was rigorously proved concerns arithmetic surfaces, thanks to tools from number theory and ergodic theory on homogeneous spaces [19,20,25].”
Restrictions of the Laplace-Beltrami eigenfunctions to submanifolds (2005, background): “One of the most popular is by describing semi-classical (Wigner) measures (see the works by Shnirelman [17], Zelditch [25], Colin de Verdière [5], Gérard-Leichtnam [7], Zelditch-Zworski [26], Helffer-MartinezRobert [9], Sarnak [16], Lindenstrauss [13] and Anantharaman [1]).”
A Dynamical Approach to Random Matrix Theory (2017, background): “One case for which QUE was rigorously proved concerns arithmetic surfaces, thanks to tools from number theory and ergodic theory on homogeneous spaces [81, 82, 100].”
Nodal Domains of Maass Forms I (2012, background): “n >0 and a k <¥ such that for any f 2C¥(X) with R X f dA =0 we have Z X jf(z)j2 f(z) dA ˝t n kf k Wk; where Wk is the Sobolev k-norm on X. Remark A.2. While QUE is known thanks to Lindenstrauss [Lin06] and Soundararajan [So10], it is not known with any rate. There is little doubt (in our minds”