The distribution of closed geodesics on the modular surface, and Duke's theorem

Einsiedler, Manfred; Lindenstrauss, Elon; Michel, Philippe; Venkatesh, Akshay

doi:10.4171/lem/58-3-2

Cited by 40 publications

(87 citation statements)

References 15 publications

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“…The point here being is that for a given center x, outside a countable set or radii μ(∂xB r ) = 0. Lemma 2.9 is a slight adaptation of Lemma 4.5 from [5]. For convenience, we added the full proof in the Appendix (see also Remark A.3).…”

Section: Maximal Entropymentioning

confidence: 99%

Equidistribution of divergent orbits and continued fraction expansion of rationals

David

Shapira

2018

J. London Math. Soc.

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We establish an equidistribution result for pushforwards of certain locally finite algebraic measures in the adelic extension of the space of lattices in the plane. As an application of our analysis, we obtain new results regarding the asymptotic normality of the continued fraction expansions of most rationals with a high denominator as well as an estimate on the length of their continued fraction expansions. By similar methods, we also establish a complementary result to Zaremba's conjecture. Namely, we show that given a bound M, for any large q, the number of rationals p/q∈[0,1] for which the coefficients of the continued fraction expansion of p/q are bounded by M is o(q1−ε) for some ε>0, which depends on M.

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Section: Maximal Entropymentioning

confidence: 99%

Equidistribution of divergent orbits and continued fraction expansion of rationals

David

Shapira

2018

J. London Math. Soc.

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“…We will show the proposition using [BK83], more precisely in the form of Lemma B.2 in [ELMV12]: There it is shown that for any small δ > 0, the entropy of µ is the limit as λ → 0 of…”

Section: Estimate Of Metric Entropy and Proof Of Theorem Amentioning

confidence: 99%

Escape of mass and entropy for diagonal flows in real rank one situations

2015

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Abstract. Let G be a connected semisimple Lie group of real rank 1 with finite center, let Γ be a non-uniform lattice in G and a any diagonalizable element in G. We investigate the relation between the metric entropy of a acting on the homogeneous space Γ\G and escape of mass. Moreover, we provide bounds on the escaping mass and, as an application, we show that the Hausdorff dimension of the set of orbits (under iteration of a) which miss a fixed open set is not full.

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“…3.42-3.45, you will appreciate the appearance of continued fractions here. Other references for the distribution of geodesics in the fundamental domain of the modular group are: Duke [144] and Einsiedler et al [154].…”

Section: Questionsmentioning

confidence: 99%

“…3, 13, 18, 23, 29, 133, 164, 186-191, 193, 195, 317, 342, 361 heat kernel, 164, 165, 186, 193 25-29, 190, 192, 193, 195 25-29, 121, 142, 188, 190, 191, 193-195 probability 54,63,79,80,150,153,154,163,196,202,207,208,243,248,256,257,267,288 …”

Section: Hintmentioning

confidence: 99%

“…I], avoiding all mention of group representations. electromagnetic spectrum, 48, 156 elementary divisor theory, 72, 79 elementary row and column operations, 79 214, 317, 345, 354, 361 fractional linear transformation, 204, 213 function, 235, 247-250, 257, 258 geometry, 112, 147, 149, 153 integral, 200, 235, 247, 248 partial differential equation, 259 point, 213, 214, 240 elliptic fractional linear transformation, 213 energy level, 48, 58, 89, 119 Epstein zeta function, 33, 44, 58, 63, 64, 67, 69, Dedekind eta function, 230, 238, 264, 374 Euclid's fifth postulate, 112, 152 Euclid's second postulate, 112 Euclidean algorithm, 204 Euclidean group, 83, 90, 136, 148, 197, 367 even integral positive matrix, 243, 256 Ewald's method of theta functions, 87 excited state, 120 expectation or mean, 26, 191, 192 F Farey fractions, 315 fast Fourier transform or FFT, 31, 46, 90, 94 Féjer kernel, 3, 34 Feynman integrals, 319 Fibonacci tiling, 98 finite analogue of Euclidean distance, 90 finite Dirichlet polygon, 225 finite Eisenstein series, 291 finite element method, 54 finite Euclidean graph, 91 finite Euclidean space, 90 finite fundamental domain, 289 finite general linear group, 222, 292, 374 finite geodesic, 222 finite horocycle, 222 finite non-Euclidean distance, 222 finite rotation group, 222 finite simple group, 242 finite symmetric space, 90 finite tessellation, 225 finite trace formula, 374, 376 finite upper half-plane, 221, 223, 227, 289, 374 finite upper half-plane graph, 223 Fischer-Griess monster group, 242 Fourier analysis on symmetric space, 9, 117, 178 Fourier analysis on the fundamental domain, 31, 39, 333, 352 Fourier coefficient, 31, 40 Fourier inversion, see inversion of a transform Fourier series, 3, 4, 6, 13, 23, 31-37, 39-41, 43, Kontorovich-Lebedev transform, 142, 164, 168-171, 175-177 Korteweg-DeVries equation, 257 Korteweg-DeVries equation, 258 Kronecker limit formula, 264, 265 Kronecker symbol, 273 Kronecker theorem, 101 Laplace operator, 110-112, 114, 115, 117-119, 133, 136 Δ, Laplace operator,5, 18, 31, 36-38, 45, 56-58, 60, 92, 155, 164, 167, 170, 173, 184, 185, 187, 258-260, 263, 267, 270, 272, 273, 280, 283, 286, 288, 297, 317, 318, 320, 321, 323, 326, 328, 329, 331, 333, 337, 338, 340-342, 347, 360, 362, 365, 366 Δ, Laplace operator, 153 Laplace series of spherical harmonics, 118 lattice space, 367 least squares, method of, 51 Lebesgue dominated convergence theorem, 6, 11, 12, 188, 195 integrable function, 2, 15, 143, 349 integrable functions, L 1 (X), 7, 11, 12, 32, 127 integral, 11, 13 measurable function, 139 measure, 2, 25, 101, 322 square integrable functions, L 2 (X), 11 left G-action,154 Legendre function P s ,113, 114, 172-177, 191, 192, 224…”

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Harmonic Analysis on Symmetric Spaces—Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane

Terras

2013

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Whenever there is a large earthquake the Earth vibrates for days afterwards. The vibrations consist of the superposition of the elastic-gravitational normal modes of the Earth that are excited by the earthquake.-From F. Gilbert [212, p. 107].A (surface or Laplace) spherical harmonic is an eigenfunction of the Laplacian on the sphere. These are the analogues of exponentials for Fourier analysis on the sphere. Laplace and Legendre introduced these functions in order to study gravitational theory in the 1780s. Spherical harmonics are necessary for the analysis of any phenomena with spherical symmetry; e.g., earthquakes, the hydrogen atom, and the solar corona. Some of these topics will be discussed later in this section.Since our treatment of harmonic analysis on the sphere is rather condensed, the reader may want to consult some of the following references for more information: Before discussing spherical harmonics, we need to understand something about the geometry of the sphere. This symmetric space is closely related to the orthogonal group O(n) of real n × n matrices U such that t UU = I, where t U denotes the transpose of U and I denotes the n × n identity matrix. The special orthogonal SO(n), is the subgroup of matrices U in O(n) such that the determinant of U is one. You can regard U in SO(n − 1) as an element of SO(n) by formingThe group O(n) is a compact group and the sphere is a compact symmetric space, making some of the analysis on it somewhat easier.Exercise 2.1.1 (The Sphere as a Quotient or Homogeneous Space). Consider the sphere S n−1 = {x ∈ R n | x = 1}. Show that S n−1 can be identified with the quotient space SO(n)/SO(n − 1), using the preceding identification of SO(n − 1) as a subgroup of SO(n). Hint.Map the coset gSO(n − 1) to the vector ge n for g in SO(n − 1) and e n = t (0,... ,0, 1).You may also want to read the discussion of the topology of spheres and orthogonal groups in Chevalley [85,. In particular, the fundamental (or Poincaré) group of SO(2) is isomorphic to Z, while that of SO(n), n ≥ 3, has order 2.From now on we shall consider only the sphere S 2 . See the references for the general case. Now the sphere S 2 is a differentiable manifold. This means that locally it looks like two-dimensional Euclidean space. To make this precise, we use the usual angular coordinates (θ , ϕ), 0 < ϕ < 2π, 0 < θ < π, pictured in Fig. 2.1, to parameterize S 2 except for the semi-circle C through the poles and (1, 0, 0). A similar coordinate patch can be constructed to cover the rest of the sphere. The equations for the rectangular coordinates (x, y, z) of a point on S 2 in terms of the angular coordinates are x = sin θ cos ϕ y = sin θ sin ϕ z = cos θ ⎫ ⎬ ⎭ (2.1)

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The distribution of closed geodesics on the modular surface, and Duke's theorem

Cited by 40 publications

References 15 publications

Equidistribution of divergent orbits and continued fraction expansion of rationals

Equidistribution of divergent orbits and continued fraction expansion of rationals

Escape of mass and entropy for diagonal flows in real rank one situations

Harmonic Analysis on Symmetric Spaces—Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane

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