Control of eigenfunctions on hyperbolic surfaces: an application of fractal uncertainty principle
Semyon Dyatlov1
1 Department of Mathematics Massachusetts Institute of Technology 77 Massachusetts Avenue Cambridge, MA 02139, USA
Journées équations aux dérivées partielles (2017), Exposé no. 4, 14 p.

This expository article, written for the proceedings of the Journées EDP (Roscoff, June 2017), presents recent work joint with Jean Bourgain [BD16] and Long Jin [DJ17]. We in particular show that eigenfunctions of the Laplacian on hyperbolic surfaces are bounded from below in L 2 norm on each nonempty open set, by a constant depending on the set but not on the eigenvalue.

Publié le :
DOI : 10.5802/jedp.654
Semyon Dyatlov 1

1 Department of Mathematics Massachusetts Institute of Technology 77 Massachusetts Avenue Cambridge, MA 02139, USA 
@incollection{JEDP_2017____A4_0,
     author = {Semyon Dyatlov},
     title = {Control of eigenfunctions on hyperbolic surfaces: an application of fractal uncertainty principle},
     booktitle = {},
     series = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     note = {talk:4},
     pages = {1--14},
     publisher = {Groupement de recherche 2434 du CNRS},
     year = {2017},
     doi = {10.5802/jedp.654},
     zbl = {1372.81101},
     language = {en},
     url = {https://proceedings.centre-mersenne.org/articles/10.5802/jedp.654/}
}
TY  - JOUR
AU  - Semyon Dyatlov
TI  - Control of eigenfunctions on hyperbolic surfaces: an application of fractal uncertainty principle
JO  - Journées équations aux dérivées partielles
N1  - talk:4
PY  - 2017
SP  - 1
EP  - 14
PB  - Groupement de recherche 2434 du CNRS
UR  - https://proceedings.centre-mersenne.org/articles/10.5802/jedp.654/
DO  - 10.5802/jedp.654
LA  - en
ID  - JEDP_2017____A4_0
ER  - 
%0 Journal Article
%A Semyon Dyatlov
%T Control of eigenfunctions on hyperbolic surfaces: an application of fractal uncertainty principle
%J Journées équations aux dérivées partielles
%Z talk:4
%D 2017
%P 1-14
%I Groupement de recherche 2434 du CNRS
%U https://proceedings.centre-mersenne.org/articles/10.5802/jedp.654/
%R 10.5802/jedp.654
%G en
%F JEDP_2017____A4_0
Semyon Dyatlov. Control of eigenfunctions on hyperbolic surfaces: an application of fractal uncertainty principle. Journées équations aux dérivées partielles (2017), Exposé no. 4, 14 p. doi : 10.5802/jedp.654. https://proceedings.centre-mersenne.org/articles/10.5802/jedp.654/

[An08] Nalini Anantharaman, Entropy and the localization of eigenfunctions, Ann. of Math. 168(2008), 435–475. | MR

[AN07] Nalini Anantharaman and Stéphane Nonnenmacher, Half-delocalization of eigenfunctions of the Laplacian on an Anosov manifold, Ann. Inst. Fourier 57(2007), 2465–2523. | MR

[BD16] Jean Bourgain and Semyon Dyatlov, Spectral gaps without the pressure condition, Ann. of Math., to appear.

[BM62] Arne Beurling and Paul Malliavin, On Fourier transforms of measures with compact support, Acta Math. 107(1962), 291–309. | MR

[Bo16] David Borthwick, Spectral theory of infinite-area hyperbolic surfaces, second edition, Birkhäuser, 2016.

[CdV85] Yves Colin de Verdière, Ergodicité et fonctions propres du Laplacien, Comm. Math. Phys. 102(1985), 497–502. | Zbl

[DJ17] Semyon Dyatlov and Long Jin, Semiclassical measures on hyperbolic surfaces have full support, preprint, . | arXiv

[Dy17] Semyon Dyatlov, Notes on fractal uncertainty principle, lecture notes in progress, http://math.mit.edu/~dyatlov/files/2017/fupnotes.pdf.

[DZ16] Semyon Dyatlov and Joshua Zahl, Spectral gaps, additive energy, and a fractal uncertainty principle, Geom. Funct. Anal. 26(2016), 1011–1094. | MR

[Ji17] Long Jin, Control for Schrödinger equation on hyperbolic surfaces, preprint, . | arXiv

[JZ17] Long Jin and Ruixiang Zhang, Fractal uncertainty principle with explicit exponent, preprint, . | arXiv

[Li06] Elon Lindenstrauss, Invariant measures and arithmetic quantum unique ergodicity, Ann. of Math. 163(2006), 165–219. | MR

[Ma06] Jens Marklof, Arithmetic quantum chaos, Encyclopedia of Mathematical Physics, Oxford: Elsevier 1(2006), 212–220.

[RS94] Zeév Rudnick and Peter Sarnak, The behaviour of eigenstates of arithmetic hyperbolic manifolds, Comm. Math. Phys. 161(1994), 195–213. | MR

[Sa11] Peter Sarnak, Recent progress on the quantum unique ergodicity conjecture, Bull. Amer. Math. Soc. 48(2011), 211–228. | MR

[Sh74] Alexander Shnirelman, Ergodic properties of eigenfunctions, Usp. Mat. Nauk. 29(1974), 181–182. | MR

[Ze87] Steve Zelditch, Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. J. 55(1987), 919–941. | MR

[Ze09] Steve Zelditch, Recent developments in mathematical quantum chaos, Curr. Dev. Math. 2009, 115–204.

[Zw12] Maciej Zworski, Semiclassical analysis, Graduate Studies in Mathematics 138, AMS, 2012. | MR | Numdam

Cité par Sources :