Quantum ergodicity. Nalini Anantharaman. 22 août Université de Strasbourg
|
|
- Jonah Mitchell
- 6 years ago
- Views:
Transcription
1 Quantum ergodicity Nalini Anantharaman Université de Strasbourg 22 août 2016
2 I. Quantum ergodicity on manifolds. II. QE on (discrete) graphs : regular graphs. III. QE on graphs : other models, perspectives.
3
4 Figure: A few eigenfunctions of the Bunimovich billiard (Heller, 89).
5 A survey of QE on manifolds M a compact riemannian manifold, of dimension d. ψ k = λ k ψ k ψ k L 2 (M) = 1 in the limit λ k +. We study the weak limits of the probability measures on M, ψ k (x) 2 dvol(x) λ k +.
6 Most results could be generalized to more general Schrödinger equations ( V (x) ) ψ (x) = E ψ (x), in the semiclassical limit 0, E E 0 R. This question is linked with the ergodic theory for the geodesic flow / billiard flow / Hamiltonian flow. Hence the name quantum ergodicity.
7 Let (ψ k ) k N be an orthonormal basis of L 2 (M), with QE theorem (simplified) : ψ k = λ k ψ k, λ k λ k+1. Theorem (Shnirelman 74, Zelditch 85, Colin de Verdière 85) Assume that the action of the geodesic flow is ergodic for the Liouville measure. Let a C 0 (M). Then 1 N(λ) λ k λ M a(x) ψ k (x) 2 dvol(x) M a(x)dvol(x) 2 0. λ
8 Equivalently, there exists a subset S N of density 1, such that a(x) ψ k (x) 2 dvol(x) a(x)dvol(x). M k S k + M
9 Equivalently, there exists a subset S N of density 1, such that a(x) ψ k (x) 2 dvol(x) a(x)dvol(x). M k S k + M Equivalently, in the weak topology. ψ k (x) 2 dvol(x) k S dvol(x) k +
10 The full statement uses analysis on phase space, i.e. T M = {(x, ξ), x M, ξ T x M}. For a = a(x, ξ) a reasonable function on T M, we can define an operator on L 2 (M), a(x, D x ) (D x = 1 i x) Say a S 0 (T M) if a is smooth and 0-homogeneous in ξ (i.e. a is a smooth fn on the sphere bundle SM).
11 A few rules of pseudodifferential calculus M = R d, a = a(x, ξ) : R d R d C. Say a S k (T R d ) if a is smooth and k-homogeneous in ξ. Define a(x, D x )u = 1 (2π) d a (x, ξ) e iξ x û(ξ)dξ. Proposition (i) If k = 0, a(x, D x ) extends to a bounded operator on L 2 (R d ) ; more generally, a(x, D x ) sends H s to H s k. (ii) [a(x, D x ), b(x, D x )] = {a, b} (x, D x ) + l.o.t. In particular for b(x, D x ) =, [ ( ), ξ a(x, Dx )] = ξ xa (x, D x ) + l.o.t.
12 For a S 0 (T M), we consider ψ k = λ k ψ k, λ k λ k+1. ψ k, a(x, D x )ψ k L 2 (M).
13 For a S 0 (T M), we consider ψ k = λ k ψ k, λ k λ k+1. ψ k, a(x, D x )ψ k L 2 (M). This amounts to M a(x) ψ k(x) 2 dvol(x) if a = a(x).
14 Let (ψ k ) k N be an orthonormal basis of L 2 (M), with QE theorem : ψ k = λ k ψ k, λ k λ k+1. Theorem (Shnirelman, Zelditch, Colin de Verdière) Assume that the action of the geodesic flow is ergodic for the Liouville measure. Let a(x, ξ) S 0 (T M). Then 1 N(λ) ψ k, a(x, D x )ψ k L 2 (M) λ k λ ξ =1 2 a(x, ξ)dxdξ 0.
15 Recall : the geodesic flow φ t : SM SM (t R) is the flow generated by the vector field ξ ξ x. Ergodicity of the geodesic flow means that the only φ t -invariant L 2 functions on SM are the constant functions. Equivalently : the only L 2 -functions a on SM such that ξ ξ xa = 0 are the constant functions.
16 Idea of proof. For any bounded operator K on L 2 (M), define the quantum variance Var λ (K) = 1 N(λ) ψ k, Kψ k L 2 2 (M). λ k λ
17 Idea of proof. For any bounded operator K on L 2 (M), define the quantum variance Var λ (K) = 1 N(λ) ψ k, Kψ k L 2 2 (M). λ k λ The proof start from the trivial observation that Var λ ([f ( ), K]) = 0 for any K, for any real function f. [f ( ), K] = f ( )K Kf ( ).
18 The proof start from the trivial observation that for any K. Var λ ([, K]) = 0 In addition, if K = a(x, D x ) is a pseudodifferential operator with a S 0 (T M), then where [, a(x, D x )] = b(x, D x ) b(x, ξ) = ( ) ξ ξ xa (x, ξ) + r(x, ξ) where r is 1-homogeneous in ξ and ξ ξ x is the derivative along the geodesic flow.
19 This implies that Var λ ( ( ξ ξ xa)(x, D x ) ) 0. λ +
20 This implies that Var λ ( ( ξ ξ xa)(x, D x ) ) 0. λ + If the geodesic flow is ergodic, this implies (density argument) Var λ (a(x, D x )) 0 λ + if a has zero mean.
21 This implies that Var λ ( ( ξ ξ xa)(x, D x ) ) 0. λ + In addition, Var λ (a(x, D x )) C ξ =1 a(x, ξ) 2 dxdξ. If the geodesic flow is ergodic, this implies (density argument) Var λ (a(x, D x )) 0 λ + if a has zero mean.
22 [Arnd Bäcker]
23 Introduction. Examples / counter-examples The Quantum Unique Ergodicity conjecture
24 Hassell s result on stadium billiards (2008) For almost every stadium, there exists a sequence of eigenfunctions that does not equidistribute over phase space.
25 The flat torus T d = R d /2πZ d Consider the special example φ k (x) = e ik.x, k Z d, φ k = k 2 φ k. These are joint eigenfunctions of, D x1,..., D xd.
26 The flat torus T d = R d /2πZ d Consider the special example φ k (x) = e ik.x, k Z d, φ k = k 2 φ k. These are joint eigenfunctions of, D x1,..., D xd. Consider the limit k +, k/ k ξ 0. One shows easily that φ k, a(x, D x )φ k L 2 (T d ) a(x, ξ 0 )dx. x T d QE does not hold for the eigenfunctions (φ k ).
27 The sphere. The eigenfunctions of S 2 are spherical harmonics, i.e. restrictions to S 2 R 3 of harmonic homogeneous polynomials in 3 variables. Harmonic homogeneous polynomials of degree l give rise to eigenfunctions of S 2 for the eigenvalue l(l + 1) (the dimension of the eigenspace is 2l + 1).
28 S 2 commutes ( with the infinitesimal rotations J 12 = 1 i x 1 x 2 x 2 x 1 ). The basis (Y m l ) l 0, m l of joint eigenfunctions of S d and J 12 cannot satisfy QE.
29 Random eigenbases (Zelditch) The set of bases of eigenfunctions of S 2 is in bijection with U(1) U(3) U(5) U(7).... Theorem (Zelditch 92, VanDerKam 97). Almost every choice of eigenbasis satisfies QE (even QUE).
30 Efns concentrating on a closed geodesic One can easily exhibit sequences of eigenfunctions concentrating on an equator. From that, Jakobson and Zelditch deduced that any convex combination of equators can be achieved as a limit of eigenfunctions.
31 Efns concentrating on a closed geodesic One can easily exhibit sequences of eigenfunctions concentrating on an equator. From that, Jakobson and Zelditch deduced that any convex combination of equators can be achieved as a limit of eigenfunctions and any invariant measure for the geodesic flow.
32 The flat torus, R d /2πZ d revisited φ λ (x) = k Z d, k 2 =λ c k e ik.x. (Jakobson, Bourgain 97, Jaffard 90, A-Fermanian-Macià, A-Macià 2012) It s not possible for a sequence of eigenfunctions to concentrate on a closed geodesic. Assume φ λ (x) 2 dx ν(dx) λ + Then ν is absolutely continuous, i.e. it has a density ν(dx) = ρ(x)dx. Moreover, for any non-empty open Ω T d, there exists c(ω) > 0 such that ν(ω) c(ω).
33 The flat torus, R d /2πZ d revisited φ λ (x) = c k e ik.x. k Z d, k 2 =λ (Jakobson, Bourgain 97, Jaffard 90, A-Fermanian-Macià, A-Macià 2012) It s not possible for a sequence of eigenfunctions to concentrate on a closed geodesic. Assume φ λ (x) 2 dx λ + ν(dx) Remark A full characterization of the set of limit measures is still missing.
34 The Quantum Unique Ergodicity conjecture QUE conjecture : Conjecture (Rudnick, Sarnak 94) On a negatively curved manifold, we have convergence of the whole sequence : ψ k, a(x, D x )ψ k L 2 (M) (x,ξ) SM a(x, ξ)dxdξ.
35 The Quantum Unique Ergodicity conjecture QUE conjecture : Conjecture (Rudnick, Sarnak 94) On a negatively curved manifold, we have convergence of the whole sequence : ψ k, a(x, D x )ψ k L 2 (M) (x,ξ) SM a(x, ξ)dxdξ. Proven by E. Lindenstrauss in the special case of arithmetic congruence surfaces, for joint eigenfunctions of the Laplacian and the Hecke operators.
36 Entropy of eigenfunctions Assume ψ k, a(x, D x )ψ k L 2 (M) (x,ξ) SM a(x, ξ)dµ(x, ξ). If M has negative curvature, A-Nonnenmacher (06) proved that µ must have positive Kolmogorov-Sinai entropy.
37 µ h KS (µ) 0 has the following properties : A measure supported by a periodic trajectory has zero entropy. According to the Ruelle-Pesin inequality, h KS (µ) d 1 λ + k dµ, k=1 where the functions λ + 1 (ρ) λ+ 2 (ρ) λ+ d 1 (ρ) > 0 are the positive Lyapunov exponents of the flow. Equality iff µ is the Liouville measure (Ledrappier-Young). On a manifold of constant curvature 1, the inequality reads h KS (µ) d 1. The functional h KS is affine on the convex set of invariant probability measures.
38 Assume for all a ψ k, a(x, D x )ψ k L 2 (M) A (06) proved that h KS (µ) > 0. (x,ξ) SM a(x, ξ)dµ(x, ξ). In constant curvature 1, A-Nonnenmacher (06) proved the explicit bound h KS (µ) d 1 2. As a corollary, the support of µ has Hausdorff dimension d.
39 Assume for all a ψ k, a(x, D x )ψ k L 2 (M) A (06) proved that h KS (µ) > 0. (x,ξ) SM a(x, ξ)dµ(x, ξ). In constant curvature 1, A-Nonnenmacher (06) proved the explicit bound h KS (µ) d 1 2. As a corollary, the support of µ has Hausdorff dimension d. Conjectured bound in variable curvature : h KS (µ) 1 2 d 1 λ + k dµ. k=1 (proven by G. Rivière for surfaces of non-positive curvature).
40 Brin and Katok s definition of entropy. For any time T > 0, introduce a distance on SM, d T (ρ, ρ ) = max t [ T /2,T /2] d(φt ρ, φ t ρ ), Denote by B T (ρ, ɛ) the ball of centre ρ and radius ɛ for the distance d T. Let µ be a φ t invariant probability measure on SM. Then, for µ-almost every ρ, the limit lim ɛ 0 lim inf 1 T + T log µ( B T (ρ, ɛ) ) def = h KS (µ, ρ) exists (local entropy). The Kolmogorov-Sinai entropy is the average of the local entropies : h KS (µ) = h KS (µ, ρ)dµ(ρ).
41 Open questions for negatively curved manifolds : Assume for all a, b, ψ k, a(x, D x )ψ k L 2 (M) M Is µ ergodic? (x,ξ) SM b(x) ψ k (x) 2 d Vol(x) ν = π µ M a(x, ξ)dµ(x, ξ). b(x)dν(x). can µ have ergodic components of zero entropy? for instance, can µ have a component carried on a closed geodesic? Is ν absolutely continuous? does it put positive mass on any open set?
42 Recent interest in local equidistribution M. Berry s semiclassical eigenfunction hypothesis. Q : let (ψ k ) be an ONB of eigenfunctions of. Let r k = r(λ k ) 0. Do we have 1 sup ψ x 0 k (x) 2 d Vol(x) 1 Vol(B(x 0, r k )) B(x 0,r k ) k 0?
43 Recent interest in local equidistribution M. Berry s semiclassical eigenfunction hypothesis. Q : let (ψ k ) be an ONB of eigenfunctions of. Let r k = r(λ k ) 0. Do we have 1 sup ψ x 0 k (x) 2 d Vol(x) 1 Vol(B(x 0, r k )) B(x 0,r k ) k 0? Hezari-Rivière, Han 2015 : negatively curved manyfolds, r k = ln λ α k, along a density 1 subsequence. Lester-Rudnick 2015, T d, d 2, r k = λ α k (α < 1 2(d 1) ), along a density 1 subsequence. Han 2015, T d, d 5, r k = λ α k eigenbases. (α < d 2 4d ), a.s. for random
Eigenvalues and eigenfunctions of the Laplacian. Andrew Hassell
Eigenvalues and eigenfunctions of the Laplacian Andrew Hassell 1 2 The setting In this talk I will consider the Laplace operator,, on various geometric spaces M. Here, M will be either a bounded Euclidean
More informationFractal uncertainty principle and quantum chaos
Fractal uncertainty principle and quantum chaos Semyon Dyatlov (UC Berkeley/MIT) July 23, 2018 Semyon Dyatlov FUP and eigenfunctions July 23, 2018 1 / 11 Overview This talk presents two recent results
More informationChaos, Quantum Mechanics and Number Theory
Chaos, Quantum Mechanics and Number Theory Peter Sarnak Mahler Lectures 2011 Hamiltonian Mechanics (x, ξ) generalized coordinates: x space coordinate, ξ phase coordinate. H(x, ξ), Hamiltonian Hamilton
More informationArithmetic quantum chaos and random wave conjecture. 9th Mathematical Physics Meeting. Goran Djankovi
Arithmetic quantum chaos and random wave conjecture 9th Mathematical Physics Meeting Goran Djankovi University of Belgrade Faculty of Mathematics 18. 9. 2017. Goran Djankovi Random wave conjecture 18.
More informationA gentle introduction to Quantum Ergodicity
A gentle introduction to Quantum Ergodicity Tuomas Sahlsten University of Bristol, UK 7 March 2017 Arithmetic study group, Durham, UK Joint work with Etienne Le Masson (Bristol) Supported by the EU: Horizon
More informationSpectra, dynamical systems, and geometry. Fields Medal Symposium, Fields Institute, Toronto. Tuesday, October 1, 2013
Spectra, dynamical systems, and geometry Fields Medal Symposium, Fields Institute, Toronto Tuesday, October 1, 2013 Dmitry Jakobson April 16, 2014 M = S 1 = R/(2πZ) - a circle. f (x) - a periodic function,
More informationRecent developments in mathematical Quantum Chaos, I
Recent developments in mathematical Quantum Chaos, I Steve Zelditch Johns Hopkins and Northwestern Harvard, November 21, 2009 Quantum chaos of eigenfunction Let {ϕ j } be an orthonormal basis of eigenfunctions
More informationQuantum Ergodicity for a Class of Mixed Systems
Quantum Ergodicity for a Class of Mixed Systems Jeffrey Galkowski University of California, Berkeley February 13, 2013 General Setup Let (M, g) be a smooth compact manifold with or without boundary. We
More informationSpectral theory, geometry and dynamical systems
Spectral theory, geometry and dynamical systems Dmitry Jakobson 8th January 2010 M is n-dimensional compact connected manifold, n 2. g is a Riemannian metric on M: for any U, V T x M, their inner product
More informationQuantum Billiards. Martin Sieber (Bristol) Postgraduate Research Conference: Mathematical Billiard and their Applications
Quantum Billiards Martin Sieber (Bristol) Postgraduate Research Conference: Mathematical Billiard and their Applications University of Bristol, June 21-24 2010 Most pictures are courtesy of Arnd Bäcker
More informationVariations on Quantum Ergodic Theorems. Michael Taylor
Notes available on my website, under Downloadable Lecture Notes 8. Seminar talks and AMS talks See also 4. Spectral theory 7. Quantum mechanics connections Basic quantization: a function on phase space
More informationWhat is quantum unique ergodicity?
What is quantum unique ergodicity? Andrew Hassell 1 Abstract A (somewhat) nontechnical presentation of the topic of quantum unique ergodicity (QUE) is attempted. I define classical ergodicity and unique
More informationQuantum symbolic dynamics
Quantum symbolic dynamics Stéphane Nonnenmacher Institut de Physique Théorique, Saclay Quantum chaos: routes to RMT and beyond Banff, 26 Feb. 2008 What do we know about chaotic eigenstates? Hamiltonian
More informationQuantum Ergodicity and Benjamini-Schramm convergence of hyperbolic surfaces
Quantum Ergodicity and Benjamini-Schramm convergence of hyperbolic surfaces Etienne Le Masson (Joint work with Tuomas Sahlsten) School of Mathematics University of Bristol, UK August 26, 2016 Hyperbolic
More informationGlobal Harmonic Analysis and the Concentration of Eigenfunctions, Part III:
Global Harmonic Analysis and the Concentration of Eigenfunctions, Part III: Improved eigenfunction estimates for the critical L p -space on manifolds of nonpositive curvature Christopher D. Sogge (Johns
More informationApplications of measure rigidity: from number theory to statistical mechanics
Applications of measure rigidity: from number theory to statistical mechanics Simons Lectures, Stony Brook October 2013 Jens Marklof University of Bristol http://www.maths.bristol.ac.uk supported by Royal
More informationErgodicity of quantum eigenfunctions in classically chaotic systems
Ergodicity of quantum eigenfunctions in classically chaotic systems Mar 1, 24 Alex Barnett barnett@cims.nyu.edu Courant Institute work in collaboration with Peter Sarnak, Courant/Princeton p.1 Classical
More informationStrichartz estimates for the Schrödinger equation on polygonal domains
estimates for the Schrödinger equation on Joint work with Matt Blair (UNM), G. Austin Ford (Northwestern U) and Sebastian Herr (U Bonn and U Düsseldorf)... With a discussion of previous work with Andrew
More informationarxiv: v2 [math.sp] 2 Sep 2015
SEMICLASSICAL ANALYSIS AND SYMMETRY REDUCTION II. EQUIVARIANT QUANTUM ERGODICITY FOR INVARIANT SCHRÖDINGER OPERATORS ON COMPACT MANIFOLDS BENJAMIN KÜSTER AND PABLO RAMACHER arxiv:508.0738v2 [math.sp] 2
More informationOn the sup-norm problem for arithmetic hyperbolic 3-manifolds
On the sup-norm problem for arithmetic hyperbolic 3-manifolds Gergely Harcos Alfréd Rényi Institute of Mathematics http://www.renyi.hu/ gharcos/ 29 September 2014 Analytic Number Theory Workshop University
More informationOrder and chaos in wave propagation
Order and chaos in wave propagation Nalini Anantharaman Université Paris 11, Orsay January 8, 2012 A physical phenomenon : harmonics (stationary vibration modes). A mathematical theorem : the spectral
More informationApplications of homogeneous dynamics: from number theory to statistical mechanics
Applications of homogeneous dynamics: from number theory to statistical mechanics A course in ten lectures ICTP Trieste, 27-31 July 2015 Jens Marklof and Andreas Strömbergsson Universities of Bristol and
More informationBouncing Ball Modes and Quantum Chaos
SIAM EVIEW Vol. 47,No. 1,pp. 43 49 c 25 Society for Industrial and Applied Mathematics Bouncing Ball Modes and Quantum Chaos Nicolas Burq Maciej Zworski Abstract. Quantum ergodicity for classically chaotic
More informationDiagonalizable flows on locally homogeneous spaces and number theory
Diagonalizable flows on locally homogeneous spaces and number theory Manfred Einsiedler and Elon Lindenstrauss Abstract.We discuss dynamical properties of actions of diagonalizable groups on locally homogeneous
More informationOptimal shape and placement of actuators or sensors for PDE models
Optimal shape and placement of actuators or sensors for PDE models Y. Privat, E. Trélat, E. uazua Univ. Paris 6 (Labo. J.-L. Lions) et Institut Universitaire de France Rutgers University - Camden, March
More informationIII. Quantum ergodicity on graphs, perspectives
III. Quantum ergodicity on graphs, perspectives Nalini Anantharaman Université de Strasbourg 24 août 2016 Yesterday we focussed on the case of large regular (discrete) graphs. Let G = (V, E) be a (q +
More informationRecent Progress on QUE
Recent Progress on QUE by Peter Sarnak Princeton University & Institute for Advanced Study In Figure 1 the domains Ω E, Ω S, Ω B are an ellipse, a stadium and a Barnett billiard table respectively. Super-imposed
More informationDiagonalizable flows on locally homogeneous spaces and number theory
Diagonalizable flows on locally homogeneous spaces and number theory Manfred Einsiedler and Elon Lindenstrauss Abstract. We discuss dynamical properties of actions of diagonalizable groups on locally homogeneous
More informationUniversity of Bristol - Explore Bristol Research. Peer reviewed version. Link to published version (if available): /
Le Masson, E., & Sahlsten,. (7). Quantum ergodicity and Benjamini- Schramm convergence of hyperbolic surfaces. Duke Mathematical Journal, 66(8), 345-346. https://doi.org/.5/794-7-7 Peer reviewed version
More informationHALF-DELOCALIZATION OF EIGENFUNCTIONS FOR THE LAPLACIAN ON AN ANOSOV MANIFOLD
HALF-DELOCALIZATION OF EIGENFUNCTIONS FOR THE LAPLACIAN ON AN ANOSOV MANIFOLD NALINI ANANTHARAMAN AND STÉPHANE NONNENMACHER Abstract We study the high-energy eigenfunctions of the Laplacian on a compact
More informationThe nodal count mystery
The nodal count mystery Peter Kuchment, Texas A&M University Partially supported by NSF 21st Midwest Geometry Conference Wichita State University, KS March 11 13 2016 Table of contents 1 Nodal patterns
More informationNON-UNIVERSALITY OF THE NAZAROV-SODIN CONSTANT
NON-UNIVERSALITY OF THE NAZAROV-SODIN CONSTANT Abstract. We prove that the Nazarov-Sodin constant, which up to a natural scaling gives the leading order growth for the expected number of nodal components
More informationMicrolocal limits of plane waves
Microlocal limits of plane waves Semyon Dyatlov University of California, Berkeley July 4, 2012 joint work with Colin Guillarmou (ENS) Semyon Dyatlov (UC Berkeley) Microlocal limits of plane waves July
More informationarxiv: v1 [nlin.cd] 27 May 2008
SOME OPEN QUESTIONS IN WAVE CHAOS arxiv:0805.4137v1 [nlin.cd] 27 May 2008 The subject area referred to as wave chaos, quantum chaos or quantum chaology has been investigated mostly by the theoretical physics
More informationQUANTUM ERGODICITY AND BEYOND. WITH A GALLERY OF PICTURES.
QUANTUM ERGODICITY AND BEYOND. WITH A GALLERY OF PICTURES. NALINI ANANTHARAMAN AND ARND BÄCKER Nalini Anantharaman completed her PhD in 2000 at Université Paris 6, under the supervision of François Ledrappier,
More information0. Introduction. Suppose that M is a compact Riemannian manifold. The Laplace operator is given in local coordinates by f = g 1/2
ASIAN J. MATH. c 2006 International Press Vol. 10, No. 1, pp. 115 126, March 2006 004 EIGENFUNCTIONS OF THE LAPLACIAN ON COMPACT RIEMANNIAN MANIFOLDS HAROLD DONNELLY Key words. Mathematical physics, quantum
More informationControl from an Interior Hypersurface
Control from an Interior Hypersurface Matthieu Léautaud École Polytechnique Joint with Jeffrey Galkowski Murramarang, microlocal analysis on the beach March, 23. 2018 Outline General questions Eigenfunctions
More informationThe Gaussian free field, Gibbs measures and NLS on planar domains
The Gaussian free field, Gibbs measures and on planar domains N. Burq, joint with L. Thomann (Nantes) and N. Tzvetkov (Cergy) Université Paris Sud, Laboratoire de Mathématiques d Orsay, CNRS UMR 8628 LAGA,
More informationThe Berry-Tabor conjecture
The Berry-Tabor conjecture Jens Marklof Abstract. One of the central observations of quantum chaology is that statistical properties of quantum spectra exhibit surprisingly universal features, which seem
More informationEstimates from below for the spectral function and for the remainder in Weyl s law
for the spectral function and for the remainder in Weyl s law D. Jakobson (McGill), jakobson@math.mcgill.ca I. Polterovich (Univ. de Montreal), iossif@dms.umontreal.ca J. (McGill), jtoth@math.mcgill.ca
More informationA geometric approach for constructing SRB measures. measures in hyperbolic dynamics
A geometric approach for constructing SRB measures in hyperbolic dynamics Pennsylvania State University Conference on Current Trends in Dynamical Systems and the Mathematical Legacy of Rufus Bowen August
More informationEntropy of C 1 -diffeomorphisms without dominated splitting
Entropy of C 1 -diffeomorphisms without dominated splitting Jérôme Buzzi (CNRS & Université Paris-Sud) joint with S. CROVISIER and T. FISHER June 15, 2017 Beyond Uniform Hyperbolicity - Provo, UT Outline
More informationMeasures on spaces of Riemannian metrics CMS meeting December 6, 2015
Measures on spaces of Riemannian metrics CMS meeting December 6, 2015 D. Jakobson (McGill), jakobson@math.mcgill.ca [CJW]: Y. Canzani, I. Wigman, DJ: arxiv:1002.0030, Jour. of Geometric Analysis, 2013
More informationIntegrable geodesic flows on the suspensions of toric automorphisms
Integrable geodesic flows on the suspensions of toric automorphisms Alexey V. BOLSINOV and Iskander A. TAIMANOV 1 Introduction and main results In this paper we resume our study of integrable geodesic
More informationGaussian Fields and Percolation
Gaussian Fields and Percolation Dmitry Beliaev Mathematical Institute University of Oxford RANDOM WAVES IN OXFORD 18 June 2018 Berry s conjecture In 1977 M. Berry conjectured that high energy eigenfunctions
More informationPOINTWISE DIMENSION AND ERGODIC DECOMPOSITIONS
POINTWISE DIMENSION AND ERGODIC DECOMPOSITIONS LUIS BARREIRA AND CHRISTIAN WOLF Abstract. We study the Hausdorff dimension and the pointwise dimension of measures that are not necessarily ergodic. In particular,
More informationSymbolic dynamics and non-uniform hyperbolicity
Symbolic dynamics and non-uniform hyperbolicity Yuri Lima UFC and Orsay June, 2017 Yuri Lima (UFC and Orsay) June, 2017 1 / 83 Lecture 1 Yuri Lima (UFC and Orsay) June, 2017 2 / 83 Part 1: Introduction
More informationPseudo-Poincaré Inequalities and Applications to Sobolev Inequalities
Pseudo-Poincaré Inequalities and Applications to Sobolev Inequalities Laurent Saloff-Coste Abstract Most smoothing procedures are via averaging. Pseudo-Poincaré inequalities give a basic L p -norm control
More informationScientific Research. Mark Pollicott
Scientific Research Mark Pollicott Most of my research has been in the general area of ergodic theory and dynamical systems, with a particular emphasis on the interaction with geometry, analysis, number
More informationGlobal Harmonic Analysis and the Concentration of Eigenfunctions, Part II:
Global Harmonic Analysis and the Concentration of Eigenfunctions, Part II: Toponogov s theorem and improved Kakeya-Nikodym estimates for eigenfunctions on manifolds of nonpositive curvature Christopher
More informationSRB measures for non-uniformly hyperbolic systems
SRB measures for non-uniformly hyperbolic systems Vaughn Climenhaga University of Maryland October 21, 2010 Joint work with Dmitry Dolgopyat and Yakov Pesin 1 and classical results Definition of SRB measure
More informationON THE PROJECTIONS OF MEASURES INVARIANT UNDER THE GEODESIC FLOW
ON THE PROJECTIONS OF MEASURES INVARIANT UNDER THE GEODESIC FLOW FRANÇOIS LEDRAPPIER AND ELON LINDENSTRAUSS 1. Introduction Let M be a compact Riemannian surface (a two-dimensional Riemannian manifold),with
More informationBOUNCING BALL MODES AND QUANTUM CHAOS
BOUNCING BALL MODES AND QUANTUM CHAOS NICOLAS BUQ AND MACIEJ ZWOSKI Abstract. Quantum ergodicity of classically chaotic systems has been studied extensively both theoretically and experimentally, in mathematics,
More informationQuantummushrooms,scars,andthe high-frequencylimitofchaoticeigenfunctions
p.1 Quantummushrooms,scars,andthe high-frequencylimitofchaoticeigenfunctions ICIAM07, July 18, 2007 Alex Barnett Mathematics Department, Dartmouth College p.2 PlanarDirichleteigenproblem Normalmodesofelasticmembraneor
More informationEntropy and the localization of eigenfunctions
Annals of Mathematics, 168 008, 435 475 Entropy and the localization of eigenfunctions By Nalini Anantharaman Abstract We study the large eigenvalue limit for the eigenfunctions of the Laplacian, on a
More informationAsymptotics of polynomials and eigenfunctions
ICM 2002 Vol. III 1 3 Asymptotics of polynomials and eigenfunctions S. Zelditch Abstract We review some recent results on asymptotic properties of polynomials of large degree, of general holomorphic sections
More informationQuantitative Oppenheim theorem. Eigenvalue spacing for rectangular billiards. Pair correlation for higher degree diagonal forms
QUANTITATIVE DISTRIBUTIONAL ASPECTS OF GENERIC DIAGONAL FORMS Quantitative Oppenheim theorem Eigenvalue spacing for rectangular billiards Pair correlation for higher degree diagonal forms EFFECTIVE ESTIMATES
More informationRECENT DEVELOPMENTS IN MATHEMATICAL QUANTUM CHAOS PRELIMINARY VERSION/COMMENTS APPRECIATED
RECENT DEVELOPMENTS IN MATHEMATICAL QUANTUM CHAOS PRELIMINARY VERSION/COMMENTS APPRECIATED STEVE ZELDITCH Abstract. This is a survey of recent results on quantum ergodicity, specifically on the large energy
More information(5) N #{j N : X j+1 X j [a, b]}
ARITHETIC QUANTU CHAOS JENS ARKLOF. Introduction The central objective in the study of quantum chaos is to characterize universal properties of quantum systems that reflect the regular or chaotic features
More informationProblems in hyperbolic dynamics
Problems in hyperbolic dynamics Current Trends in Dynamical Systems and the Mathematical Legacy of Rufus Bowen Vancouver july 31st august 4th 2017 Notes by Y. Coudène, S. Crovisier and T. Fisher 1 Zeta
More informationADELIC DYNAMICS AND ARITHMETIC QUANTUM UNIQUE ERGODICITY
ADELIC DYNAMICS AND ARITHMETIC QUANTUM UNIQUE ERGODICITY ELON LINDENSTRAUSS 1. Introduction Let M be a complete Riemannian manifold with finite volume which we initially assume to be compact. Then since
More informationSmooth Ergodic Theory and Nonuniformly Hyperbolic Dynamics
CHAPTER 2 Smooth Ergodic Theory and Nonuniformly Hyperbolic Dynamics Luis Barreira Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal E-mail: barreira@math.ist.utl.pt url:
More informationLocal smoothing and Strichartz estimates for manifolds with degenerate hyperbolic trapping
Local smoothing and Strichartz estimates for manifolds with degenerate hyperbolic trapping H. Christianson partly joint work with J. Wunsch (Northwestern) Department of Mathematics University of North
More informationSYMBOLIC DYNAMICS FOR HYPERBOLIC SYSTEMS. 1. Introduction (30min) We want to find simple models for uniformly hyperbolic systems, such as for:
SYMBOLIC DYNAMICS FOR HYPERBOLIC SYSTEMS YURI LIMA 1. Introduction (30min) We want to find simple models for uniformly hyperbolic systems, such as for: [ ] 2 1 Hyperbolic toral automorphisms, e.g. f A
More informationFocal points and sup-norms of eigenfunctions
Focal points and sup-norms of eigenfunctions Chris Sogge (Johns Hopkins University) Joint work with Steve Zelditch (Northwestern University)) Chris Sogge Focal points and sup-norms of eigenfunctions 1
More informationRecent Progress on the Quantum Unique Ergodicity Conjecture
Recent Progress on the Quantum Unique Ergodicity Conjecture by Peter Sarnak Abstract We report on some recent striking advances on the quantum unique ergodicity or QUE conjecture, concerning the distribution
More informationTHE SEMICLASSICAL THEORY OF DISCONTINUOUS SYSTEMS AND RAY-SPLITTING BILLIARDS
THE SEMICLASSICAL THEORY OF DISCONTINUOUS SYSTEMS AND RAY-SPLITTING BILLIARDS DMITRY JAKOBSON, YURI SAFAROV, AND ALEXANDER STROHMAIER Abstract. We analyze the semiclassical limit of spectral theory on
More informationSPECTRAL THEOREM FOR COMPACT SELF-ADJOINT OPERATORS
SPECTRAL THEOREM FOR COMPACT SELF-ADJOINT OPERATORS G. RAMESH Contents Introduction 1 1. Bounded Operators 1 1.3. Examples 3 2. Compact Operators 5 2.1. Properties 6 3. The Spectral Theorem 9 3.3. Self-adjoint
More informationKUIPER S THEOREM ON CONFORMALLY FLAT MANIFOLDS
KUIPER S THEOREM ON CONFORMALLY FLAT MANIFOLDS RALPH HOWARD DEPARTMENT OF MATHEMATICS UNIVERSITY OF SOUTH CAROLINA COLUMBIA, S.C. 29208, USA HOWARD@MATH.SC.EDU 1. Introduction These are notes to that show
More informationUnique equilibrium states for geodesic flows in nonpositive curvature
Unique equilibrium states for geodesic flows in nonpositive curvature Todd Fisher Department of Mathematics Brigham Young University Fractal Geometry, Hyperbolic Dynamics and Thermodynamical Formalism
More informationThe first half century of entropy: the most glorious number in dynamics
The first half century of entropy: the most glorious number in dynamics A. Katok Penn State University This is an expanded version of the invited talk given on June 17, 2003 in Moscow at the conference
More informationArithmetic Quantum Unique Ergodicity for Γ \H
M.E. & T.W. Arithmetic Quantum Unique Ergodicity for Γ \H February 4, 200 Contents Introduction to (A)QUE.................................... (Arithmetic) Quantum Unique Ergodicity.................. 2.2
More informationQUANTUM ERGODICITY AND MIXING OF EIGENFUNCTIONS
QUANTUM ERGODICITY AND MIXING OF EIGENFUNCTIONS STEVE ZELDITCH Abstract. This article surveys mathematically rigorous results on quantum ergodic and mixing systems, with an emphasis on general results
More informationarxiv: v1 [math-ph] 29 Oct 2018
SMALL SCALE QUANTUM ERGODICITY IN CAT MAPS. I XIAOLONG HAN arxiv:80.949v [math-ph] 29 Oct 208 Abstract. In this series, we investigate quantum ergodicity at small scales for linear hyperbolic maps of the
More informationErgodic billiards that are not quantum unique ergodic
ANNALS OF MATHEMATICS Ergodic billiards that are not quantum unique ergodic By Andrew Hassell With an Appendix by Andrew Hassell and Luc Hillairet SECOND SERIES, VOL. 171, NO. 1 January, 2010 anmaah Annals
More informationCounting geodesic arcs in a fixed conjugacy class on negatively curved surfaces with boundary
Counting geodesic arcs in a fixed conjugacy class on negatively curved surfaces with boundary Mark Pollicott Abstract We show how to derive an asymptotic estimates for the number of closed arcs γ on a
More informationDispersive Equations and Hyperbolic Orbits
Dispersive Equations and Hyperbolic Orbits H. Christianson Department of Mathematics University of California, Berkeley 4/16/07 The Johns Hopkins University Outline 1 Introduction 3 Applications 2 Main
More informationFractal Weyl Laws and Wave Decay for General Trapping
Fractal Weyl Laws and Wave Decay for General Trapping Jeffrey Galkowski McGill University July 26, 2017 Joint w/ Semyon Dyatlov The Plan The setting and a brief review of scattering resonances Heuristic
More informationINFINITESIMAL LYAPUNOV FUNCTIONS, INVARIANT CONE FAMILIES AND STOCHASTIC PROPERTIES OF SMOOTH DYNAMICAL SYSTEMS. Anatole KATOK 1
INFINITESIMAL LYAPUNOV FUNCTIONS, INVARIANT CONE FAMILIES AND STOCHASTIC PROPERTIES OF SMOOTH DYNAMICAL SYSTEMS by Anatole KATOK 1 in collaboration with Keith BURNS 2 Abstract. We establish general criteria
More informationInégalités de dispersion via le semi-groupe de la chaleur
Inégalités de dispersion via le semi-groupe de la chaleur Valentin Samoyeau, Advisor: Frédéric Bernicot. Laboratoire de Mathématiques Jean Leray, Université de Nantes January 28, 2016 1 Introduction Schrödinger
More informationProbabilistic Sobolev embeddings, applications to eigenfunctions estimates
Contemporary Mathematics Probabilistic Sobolev embeddings, applications to eigenfunctions estimates Nicolas Bur and Gilles Lebeau Abstract. The purpose of this article is to present some probabilistic
More informationLuminy Lecture 2: Spectral rigidity of the ellipse joint work with Hamid Hezari. Luminy Lecture April 12, 2015
Luminy Lecture 2: Spectral rigidity of the ellipse joint work with Hamid Hezari Luminy Lecture April 12, 2015 Isospectral rigidity of the ellipse The purpose of this lecture is to prove that ellipses are
More informationarxiv: v2 [math.sp] 1 Jun 2016
QUANUM ERGOICIY AN BENJAMINI-SCHRAMM CONVERGENCE OF HYPERBOLIC SURFACES EIENNE LE MASSON AN UOMAS SAHLSEN arxiv:65.57v [math.sp] Jun 6 Abstract. We present a quantum ergodicity theorem for fixed spectral
More informationFocal points and sup-norms of eigenfunctions
Focal points and sup-norms of eigenfunctions Chris Sogge (Johns Hopkins University) Joint work with Steve Zelditch (Northwestern University)) Chris Sogge Focal points and sup-norms of eigenfunctions 1
More informationShort note on compact operators - Monday 24 th March, Sylvester Eriksson-Bique
Short note on compact operators - Monday 24 th March, 2014 Sylvester Eriksson-Bique 1 Introduction In this note I will give a short outline about the structure theory of compact operators. I restrict attention
More informationLECTURE 15: COMPLETENESS AND CONVEXITY
LECTURE 15: COMPLETENESS AND CONVEXITY 1. The Hopf-Rinow Theorem Recall that a Riemannian manifold (M, g) is called geodesically complete if the maximal defining interval of any geodesic is R. On the other
More informationQuantum decay rates in chaotic scattering
Quantum decay rates in chaotic scattering S. Nonnenmacher (Saclay) + M. Zworski (Berkeley) National AMS Meeting, New Orleans, January 2007 A resonant state for the partially open stadium billiard, computed
More informationAnalysis in weighted spaces : preliminary version
Analysis in weighted spaces : preliminary version Frank Pacard To cite this version: Frank Pacard. Analysis in weighted spaces : preliminary version. 3rd cycle. Téhéran (Iran, 2006, pp.75.
More informationOPEN PROBLEMS IN DYNAMICS AND RELATED FIELDS
OPEN PROBLEMS IN DYNAMICS AND RELATED FIELDS ALEXANDER GORODNIK Contents 1. Local rigidity 2 2. Global rigidity 3 3. Measure rigidity 5 4. Equidistribution 8 5. Divergent trajectories 12 6. Symbolic coding
More informationNotes on fractal uncertainty principle version 0.5 (October 4, 2017) Semyon Dyatlov
Notes on fractal uncertainty principle version 0.5 (October 4, 2017) Semyon Dyatlov E-mail address: dyatlov@math.mit.edu Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts
More informationA non-uniform Bowen s equation and connections to multifractal analysis
A non-uniform Bowen s equation and connections to multifractal analysis Vaughn Climenhaga Penn State November 1, 2009 1 Introduction and classical results Hausdorff dimension via local dimensional characteristics
More informationESI Lectures on Semiclassical Analysis and Quantum Ergodicity 2012
ESI Lectures on Semiclassical Analysis and Quantum Ergodicity 2012 Andrew Hassell Department of Mathematics Australian National University ESI, Vienna, July August 2012 Outline Introduction Semiclassical
More informationArithmetic Quantum Unique Ergodicity for Symplectic Linear Maps of the Multidimensional Torus
Arithmetic Quantum Unique Ergodicity for Symplectic Linear Maps of the Multidimensional Torus Thesis submitted for the degree Doctor of Philosophy by Dubi Kelmer Prepared under the supervision of Professor
More informationCALCULUS ON MANIFOLDS. 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M =
CALCULUS ON MANIFOLDS 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M = a M T am, called the tangent bundle, is itself a smooth manifold, dim T M = 2n. Example 1.
More informationTransport Continuity Property
On Riemannian manifolds satisfying the Transport Continuity Property Université de Nice - Sophia Antipolis (Joint work with A. Figalli and C. Villani) I. Statement of the problem Optimal transport on Riemannian
More informationPARTIAL HYPERBOLICITY, LYAPUNOV EXPONENTS AND STABLE ERGODICITY
PARTIAL HYPERBOLICITY, LYAPUNOV EXPONENTS AND STABLE ERGODICITY K. BURNS, D. DOLGOPYAT, YA. PESIN Abstract. We present some results and open problems about stable ergodicity of partially hyperbolic diffeomorphisms
More informationAn introduction to Birkhoff normal form
An introduction to Birkhoff normal form Dario Bambusi Dipartimento di Matematica, Universitá di Milano via Saldini 50, 0133 Milano (Italy) 19.11.14 1 Introduction The aim of this note is to present an
More informationABSOLUTE CONTINUITY OF FOLIATIONS
ABSOLUTE CONTINUITY OF FOLIATIONS C. PUGH, M. VIANA, A. WILKINSON 1. Introduction In what follows, U is an open neighborhood in a compact Riemannian manifold M, and F is a local foliation of U. By this
More informationPoincaré Inequalities and Moment Maps
Tel-Aviv University Analysis Seminar at the Technion, Haifa, March 2012 Poincaré-type inequalities Poincaré-type inequalities (in this lecture): Bounds for the variance of a function in terms of the gradient.
More informationGEOMETRIC QUANTIZATION
GEOMETRIC QUANTIZATION 1. The basic idea The setting of the Hamiltonian version of classical (Newtonian) mechanics is the phase space (position and momentum), which is a symplectic manifold. The typical
More information