## Eigenvalue Spacings for Quantized Cat Maps

Citations: | 1 - 0 self |

### BibTeX

@MISC{Gamburd_eigenvaluespacings,

author = {Alex Gamburd and John Lafferty and Dan Rockmore},

title = {Eigenvalue Spacings for Quantized Cat Maps},

year = {}

}

### OpenURL

### Abstract

According to one of the basic conjectures in Quantum Chaos, the eigenvalues of a quantized chaotic Hamiltonian behave like the spectrum of the typical member of the appropriate ensemble of random matrices. We study one of the simplest examples of this phenomenon in the context of ergodic actions of groups generated by several linear toral automorphisms { \cat maps". Our numerical experiments indicate that for \generic" choices of cat maps, the unfolded consecutive spacings distribution in the irreducible components of the N-th quantization (given by the N-dimensional Weil representation) approaches the GOE/GSE law of Random Matrix Theory. For certain special \arithmetic " transformations, related to the Ramanujan graphs of Lubotzky, Phillips and Sarnak, the experiments indicate that the unfolded consecutive spacings distribution follows Poisson statistics; we provide a sharp estimate in that direction.

### Citations

1843 | Random Graphs
- Bollobás
- 2001
(Show Context)
Citation Context ...umerical experiments in [19] indicate that the spacing distribution of eigenvalues of random regular graphs follows GOE distribution. Since random regular graphs asymptotically have logarithmic girth =-=[9]-=-, the argument outlined above also applies to give similar discrepancy bound for the spectrum of random regular graphs. 5.3. Proof of Theorem 2. We now consider Ramanujan elements z p ; the associated... |

228 |
Ergodic problems of classical mechanics
- Arnold, Avez
- 1967
(Show Context)
Citation Context ...FOSR. 1 2 ALEX GAMBURD, JOHN LAFFERTY, AND DAN ROCKMORE which have received a lot of attention in the physics and mathematics literature, go by the name \cat maps," which derives from the picture=-=s in [2]-=- which show a cartoon cat face and its images under a few iterates of A, displaying the chaotic features of x 7! Ax. The quantization of such a linear transformation can be carried out by periodizing ... |

141 |
Random Matrices, 2nd ed
- Mehta
- 1991
(Show Context)
Citation Context ...numerical experiments, described in Section 4 indicate that the unfolded consecutive spacing distribution for \generic" z (see the discussion in Section 4) follows the GSE law of Random Matrix Th=-=eory [37-=-] for U q (z) and GOE law of Random Matrix Theory for U + q (z). We also consider certain arithmetic, or Ramanujan elements introduced by Lubotzky, Phillips and Sarnak [32], dened as follows. Let H(Z)... |

124 |
Symmetric random walks on groups
- Kesten
- 1959
(Show Context)
Citation Context ... with a similar expression for U q (z): (4) q (z) = 4 q 1 (q 1)=4 X j=0 j (U q (z)) : It is not dicult to show [38, 41, 15] that these converge to the measure k (t),srst considered by Kesten in [24], which is supported in the interval [ 2 p 2k 1; 2 p 2k 1] and given by (5) d k (t) = p 2k 1 t 2 =4 2k(1 (t=2k) 2 ) dt: Our numerical experiments, described in Section 4 indicate that the unfolded c... |

118 |
Fourier analysis on finite groups and applications
- Terras
- 1999
(Show Context)
Citation Context ... larger discrepancy than the random elements. We now turn to the proof of these results. 5. Discrepancy 5.1. Trace formula for regular graphs. We begin by reviewing the basic denitions, referring to [=-=11, 45]-=- for details. Let X = (V; E) be a k-regular graph, that is a graph with each vertex having k neighbors. The adjacency EIGENVALUE SPACINGS FOR QUANTIZED CAT MAPS 11 matrix of X, A(X) is the j V j by j ... |

102 |
The Selberg trace formula for PSL(2
- Hejhal
- 1976
(Show Context)
Citation Context ...see Figure 3). We conclude section 5 by proving the following sharp lower bound, which is the analogue of the lower bounds for the remainder term in Weyl's law for arithmetic hyperbolic surfaces, see =-=-=-[18], [35], and [15]. Theorem 2. Fix p 3, let X q;p denote the Cayley graph of SL 2 (F q ) associated with the Ramanujan element z p . Let k = 1 2 (p + 1). Then D(Xq;p ; k ) 1 q log 2 q = 1 j X p;q... |

76 |
Characterization of chaotic quantum spectra and universality of level ° uctuation
- Bohigas, Giannoni, et al.
- 1984
(Show Context)
Citation Context ...stems with few degrees of freedom with chaotic classical dynamics; in fact, RMT lies at the heart of one of the basic conjectures in Quantum Chaos. Formulated by Bohigas, Giannoni, and Schmit in 1984 =-=[8]-=-, it asserts that the eigenvalues of a quantized chaotic Hamiltonian (after suitable unfolding) behave like the spectrum of a typical member of the appropriate ensemble of random matrices. This conjec... |

60 |
The expected eigenvalue distribution of a large regular graph
- McKay
- 1981
(Show Context)
Citation Context ...here α(z) = max L∈ supp(z) �L�. Here the norm of a matrix L is defined by �Lx� �L� =sup x�=0 �x� and the norm of x = (x1,x2) is given by �x� = �q � − 1 (20) 2 � x 2 1 + x2 2 . By theorem 1.1 of McKay =-=[36]-=-, the bound on girth (20)implies the convergence to Kesten measure. Now let and F(Xq(z), t) = F2k(t) = � t � t −2 √ 2k−1 −2 √ 2k−1 ν2k(x) dx. µXq(z)(x) dx By theorem 4.4 of McKay [36], for every t we ... |

58 |
Some applications of modular forms
- Sarnak
- 1990
(Show Context)
Citation Context ...iables, dened by Q(x 0 ; x 1 ; x 2 ; x 3 ) = x 2 0 + 4q 2 (x 2 1 + x 2 2 + x 2 3 ): The optimal estimate for s Q (p n ), is obtained by appealing to Ramanujan bounds proved by Eichler and Igusa (see [=-=-=-39]); the following bound, which 14 ALEX GAMBURD, JOHN LAFFERTY, AND DAN ROCKMORE suces for our purposes, can be obtained by elementary means (see [11]): (29) s Q (p n ) = O p n(1+) q 3 + p n=2(1+2) ... |

53 |
Arithmetic quantum chaos
- Sarnak
- 1995
(Show Context)
Citation Context ... in[15]. One way in which the difference between RMT and the Poisson distribution manifests itself is in the speed of convergence to the Kesten measure: the convergence is much faster in the RMT case =-=[41, 15]-=-. Our numerical experiments, detailed in section 4,indicate that for a fixed z all irreducible representations of SL2(Fq) behave in a similar way to U − q and U + q (depending on parity) with respect ... |

46 |
Amenability, Kazhdan’s property T, strong ergodicity and invariant means for ergodic group-actions
- Schmidt
- 1981
(Show Context)
Citation Context ...A1, ...,Ak with Ai ∈ SL(2, Z). Theaction of the group Ɣ is strongly ergodic if the associated element in the group ring of SL(2, Z), zA1,...,Ak = A1 + A −1 1 + ···+ Ak + A −1 k (1) has a spectral gap =-=[44]-=-. Let supp(z) ={A1, ...,Ak} and Ɣz be the group generated by supp(z). Numerical experiments [29, 30] indicate that a ‘generic’ element z in the group ring of SL(2, Z) has a spectral gap. In [14] it is... |

44 | Classical limit of the quantized hyperbolic toral automorphisms
- Esposti, Gra–, et al.
- 1995
(Show Context)
Citation Context ...on can be carried out by periodizing any one of the standard quantization procedures in R 2 . This has been carried outsrst by Hannay and Berry in [16] and has since been studied by many authors, see =-=[3, 5, 12, 13, 22, 25] and-=- references therein. We will adopt the quantization procedure given by Kurlberg and Rudnick in [27]. It yields for each integer N 1 (\N = 1=~") a unitary matrix UN (A) acting on L 2 (Z=NZ). As w... |

43 |
Répartition asymptotique des valeurs propres de l’opérateur de Hecke Tp
- Serre
- 1997
(Show Context)
Citation Context ...U q +1 + q (z)) (3) j=1 which is a probability measure supported in [−2k,2k]with a similar expression for U − q (z): µ − (q−1)/4 4 � q (z) = δλj (U q − 1 − q (z)). (4) j=1 It is not difficult to show =-=[39, 42, 15]-=- that these converge to the measure νk(t),firstconsidered by Kesten in [24], which is supported in the interval [−2 √ 2k − 1, 2 √ 2k − 1] and given by � 2k − 1 − t2 /4 dνk(t) = 2πk(1 − (t/2k) 2 dt. (5... |

38 | Hecke theory and equidistribution for the quantization of linear maps of the torus, preprint chao-dyn/9901031
- Kurlberg, Rudnick
(Show Context)
Citation Context ...st by Hannay and Berry in [16] and has since been studied by many authors, see [3, 5, 12, 13, 22, 25] and references therein. We will adopt the quantization procedure given by Kurlberg and Rudnick in =-=[27]. It-=- yields for each integer N 1 (\N = 1=~") a unitary matrix UN (A) acting on L 2 (Z=NZ). As we review in Section 2, and as detailed in [27], UN (A) is essentially the Weil or metaplectic represent... |

33 |
Cayley graphs: Eigenvalues, expanders and random walks
- Lubotzky
- 1995
(Show Context)
Citation Context ...nt z in the group ring of SL(2; Z) has a spectral gap. In [14] it is proved that z has a spectral gap if the Hausdor dimension of the limit set of z is large enough; see [44] for related results and [=-=33, 34]-=- for the discussion of this problem. 1 The classical limit can be thought of as a random walk supported on the toral automorphisms in question, or, following Arnold and Krylov [1], as a dynamical syst... |

29 | Index and dynamics of quantized contact transformations, Ann. Inst. Fourier (Grenoble) 47
- Zelditch
- 1997
(Show Context)
Citation Context ...tant negative curvature, following the pioneering numerical experiments in [43, 7, 10]. An important model for understanding quantization of classically chaotic systems is afforded by symplectic maps =-=[48]-=-. The simplest of these are the linear area-preserving transformations of the torus T 2 = R2 /Z2 ; that is, transformations � � � � x1 x1 ↦→ A with x2 x2 A ∈ SL(2, Z). These transformations, which hav... |

28 |
The cat maps : Quantum mechanics and Classical motion, Nonlinearity 4
- Keating
- 1991
(Show Context)
Citation Context ... breakthroughs by Kurlberg and Rudnick [27]. The distribution of the eigenvalues of UN(A) is degenerate, and not what is expected for the quantization of a generic chaotic system, as shown by Keating =-=[20]-=-. Following an early attempt at restoring generic statistics by Lakshminarayan and Balazs in [31], several ways of recovering the predicted random matrix distribution for modified cat maps have been p... |

26 |
Esposti, Quantization of the orientation preserving automorphisms of the torus
- Degli
- 1993
(Show Context)
Citation Context ...on can be carried out by periodizing any one of the standard quantization procedures in R 2 . This has been carried outsrst by Hannay and Berry in [16] and has since been studied by many authors, see =-=[3, 5, 12, 13, 22, 25] and-=- references therein. We will adopt the quantization procedure given by Kurlberg and Rudnick in [27]. It yields for each integer N 1 (\N = 1=~") a unitary matrix UN (A) acting on L 2 (Z=NZ). As w... |

24 |
Random Matrices in Physics
- Wigner
- 1967
(Show Context)
Citation Context ... Wigner’s suggestion in the early fifties that the resonance lines of heavy nuclei, their determination by analytic means being intractable, might be modelled by the spectrum of a large random matrix =-=[47]-=-. While it was conceived of as a statistical approach to systems with many degrees of freedom, RMTalsoapplies to systems with few degrees of freedom with chaotic classical dynamics; in fact, RMT lies ... |

22 | Fast Fourier analysis for SL2 over a finite field and related numerical experiments, Experimental Mathematics 1
- Lafferty, Rockmore
- 1992
(Show Context)
Citation Context ... = Uq(I − S(q))/2. (16)sEigenvalue spacings for quantized cat maps 3493 4. Numerical experiments In this section we review the relevant representation theory of SL2(Fq), following the presentation of =-=[28]-=-, and then describe our numerical experiments, which compare the eigenvalue density and spacing distribution of the Weil representations for generic and Ramanujan elements. The irreducible representat... |

22 | Groups and expanders
- Lubotzky, Weiss
- 1993
(Show Context)
Citation Context ... z in the group ring of SL(2, Z) has a spectral gap. In [14] it is proved that z has a spectral gap if the Hausdorff dimension of the limit set of Ɣz is large enough; see [45] for related results and =-=[33, 34]-=- for the discussion of this problem. We consider the quantizations Uq(z), � � � � −1 −1 Uq(z) = Uq(A1) + Uq A1 + ···+ Uq(Ak) + Uq Ak . For technical reasons, detailed below, we restrict ourselves to p... |

21 |
Arithmetical chaos and violation of universality in energy level statistics
- Bolte, Steil, et al.
- 1992
(Show Context)
Citation Context ...K 3487s3488 AGamburdet al the Poisson distribution; the Poisson distribution is also expected for arithmetic surfaces of constant negative curvature, following the pioneering numerical experiments in =-=[43, 7, 10]-=-. An important model for understanding quantization of classically chaotic systems is afforded by symplectic maps [48]. The simplest of these are the linear area-preserving transformations of the toru... |

20 | Quantization of linear maps on a torus-Fresnel diffraction by a periodic grating, Phys - Hannay, Berry - 1980 |

20 |
Random matrices, 2nd edn
- Mehta
- 1991
(Show Context)
Citation Context ...umerical experiments, described in section 4, indicate that the unfolded consecutive spacing distribution for ‘generic’ z (see the discussion in section 4) follows the GSE law of random matrix theory =-=[37]-=- for U − q (z) and GOE law of random matrix theory for U + q (z). We also consider certain arithmetic, or Ramanujan elements introduced by Lubotzky, Phillips and Sarnak [32], defined as follows. Let H... |

20 | Spectra of elements in the group ring of SU(2
- Gamburd, Jakobson, et al.
- 1999
(Show Context)
Citation Context ...wo-spinor processing in a magneticseld. In this paper we show how to recover the RMT predictions while staying within the framework of linear maps and representation theory. The basic idea, following =-=[15], is -=-to consider the ergodic action of the group generated by several 1 linear toral automorphisms, i.e., \several maps of a cat". More precisely, let A1 ;:::;A k be the group generated by the transfo... |

18 |
P.: Number variance for arithmetic hyperbolic surfaces
- Luo
(Show Context)
Citation Context ...see figure 3). We conclude section 5 by proving the following sharp lower bound, which is the analogue of the lower bounds for the remainder term in Weyl’s law for arithmetic hyperbolic surfaces, see =-=[18, 35, 15]-=-. Theorem 2. Fix p � 3, letXq,p denote the Cayley graph of SL2(Fq) associated with the Ramanujan element zp. Letk = 1 2 (p +1). Then D � µXq,p ,νk � 1 ≫ q log2 q = 1 2. Quantum mechanics on the torus ... |

18 |
Chaotic billiards generated by arithmetic groups
- Bogomolny, Georgeot, et al.
- 1992
(Show Context)
Citation Context ...integrable systems follow the Poisson distribution; the Poisson distribution is also expected for arithmetic surfaces of constant negative curvature, following the pioneering numerical experiments in =-=[42, 7, 10-=-]. An important model for understanding quantization of classically chaotic systems is aorded by symplectic maps [47]. The simplest of these are the linear area-preserving transformations of the torus... |

15 |
An Introduction to the Theory of Numbers. 5th edn
- Hardy, Wright
- 1989
(Show Context)
Citation Context ...ilton quaternions α = x0 + x1i + x2j + x3k,xj ∈ Z. Let¯α = x0 − x1i − x2j − x3k and N(α) = α ¯α. Forp�3a prime number let ˜g 1, ˜g 2, ..., ˜g k be a subset of S ={α ∈ H(Z)|N(α) = p} (it is well known =-=[17]-=- that S has 8(p +1) elements) satisfying (1) ˜g j1 �= ε ˜g j2 for j1 �= j2 and ε ∈{±1, ±i, ±j,±k} a unit (2) ˜g j1 �= ε ˜g j2 for any j1,j2 and ε a unit. Among these there are p +1elements with x0 > 0... |

15 |
Statistical properties of eigenvalues of the Hecke operators, in: Analytic Number Theory and Diophantine Problems
- Sarnak
- 1984
(Show Context)
Citation Context ...U q +1 + q (z)) (3) j=1 which is a probability measure supported in [−2k,2k]with a similar expression for U − q (z): µ − (q−1)/4 4 � q (z) = δλj (U q − 1 − q (z)). (4) j=1 It is not difficult to show =-=[39, 42, 15]-=- that these converge to the measure νk(t),firstconsidered by Kesten in [24], which is supported in the interval [−2 √ 2k − 1, 2 √ 2k − 1] and given by � 2k − 1 − t2 /4 dνk(t) = 2πk(1 − (t/2k) 2 dt. (5... |

11 |
The behaviour of general theta functions under the modular group and the characters of binary modular congruence groups
- Kloosterman
- 1946
(Show Context)
Citation Context ...tary matrix UN(A) acting on L2 (Z/NZ). Aswereviewinsection 2, andasdetailed in [27], UN(A) is essentially the Weil or metaplectic representation of A reduced modulo N (first considered by Kloosterman =-=[26]-=-). The behaviour of the eigenstates of UN(A) has been the subject of intensive investigations in the papers cited above, with important recent breakthroughs by Kurlberg and Rudnick [27]. The distribut... |

11 |
Uniform distribution of points on a sphere and some ergodic properties of solutions of linear ordinary differential equations in the complex plane
- Arnold, Kry1ov
- 1962
(Show Context)
Citation Context ...ted results and [33, 34] for the discussion of this problem. 1 The classical limit can be thought of as a random walk supported on the toral automorphisms in question, or, following Arnold and Krylov =-=[1]-=-, as a dynamical system with noncommutative time. EIGENVALUE SPACINGS FOR QUANTIZED CAT MAPS 3 We consider the quantizations U q (z), U q (z) = U q (A 1 ) + U q (A 1 1 ) + : : : + U q (A k ) + U q (A ... |

10 | Eigenvalue spacings for regular graphs
- Jakobson, Miller, et al.
- 2003
(Show Context)
Citation Context ...r every t we have jF (X q (z); t) F 2k (t)js24e p 2k 1 2 (girth(X q (z)) + 2) : This estimate, combined with the bound (20) completes the proof of Theorem 1. We remark that numerical experiments in [=-=19]-=- indicate that the spacing distribution of eigenvalues of random regular graphs follows GOE distribution. Since random regular graphs asymptotically have logarithmic girth [9], the argument outlined a... |

9 | Berry: Quantization of linear maps on a torus — Fresnel diffraction by periodic grating - Hannay, V - 1980 |

8 |
Valette: Elementary Number Theory, Group Theory, and Ramanujan Graphs. Cambridge U
- Davidoff, Sarnak, et al.
- 2003
(Show Context)
Citation Context ...g larger discrepancy than therandom elements. We now turn to the proof of these results. 5. Discrepancy 5.1. Trace formula for regular graphs We begin by reviewing the basic definitions, referring to =-=[11, 46]-=- for details. Let X = (V, E) be a k-regular graph, that is a graph with each vertex having k neighbours. The adjacency matrix of X, A(X) is the |V | by |V | matrix, with rows and columns indexed by ve... |

8 |
Quantum boundary conditions for torus maps, Nonlinearity 12
- Keating, Mezzadri, et al.
- 1999
(Show Context)
Citation Context ...ion can be carried out by periodizing any one of the standard quantization procedures in R 2 .Thishasbeen carried out first by Hannay and Berry in [16] and has since been studied by many authors; see =-=[3, 5, 12, 13, 22, 25]-=- and references therein. We will adopt the quantization procedure given by Kurlberg and Rudnick in [27]. It yields for each integer N � 1(‘N = 1/¯h’) a unitary matrix UN(A) acting on L2 (Z/NZ). Aswere... |

8 | Numerical investigation of the spectrum for certain families of Cayley graphs
- Lafferty, Rockmore
- 1993
(Show Context)
Citation Context ...ement in the group ring of SL(2, Z), zA1,...,Ak = A1 + A −1 1 + ···+ Ak + A −1 k (1) has a spectral gap [44]. Let supp(z) ={A1, ...,Ak} and Ɣz be the group generated by supp(z). Numerical experiments =-=[29, 30]-=- indicate that a ‘generic’ element z in the group ring of SL(2, Z) has a spectral gap. In [14] it is proved that z has a spectral gap if the Hausdorff dimension of the limit set of Ɣz is large enough;... |

8 |
Degli Esposti and R. Giachetti, Quantization of a class of piecewise affine transformations on the torus
- Bièvre, M
- 1995
(Show Context)
Citation Context ...on can be carried out by periodizing any one of the standard quantization procedures in R 2 . This has been carried outsrst by Hannay and Berry in [16] and has since been studied by many authors, see =-=[3, 5, 12, 13, 22, 25] and-=- references therein. We will adopt the quantization procedure given by Kurlberg and Rudnick in [27]. It yields for each integer N 1 (\N = 1=~") a unitary matrix UN (A) acting on L 2 (Z=NZ). As w... |

7 |
I and Avez A 1968 Ergodic problems of classical mechanics The
- Arnold
(Show Context)
Citation Context ...→ A with x2 x2 A ∈ SL(2, Z). These transformations, which have received considerable attention in the physics and mathematics literature, go by the name ‘cat maps’, which derives from the pictures in =-=[2]-=- that show a cartoon cat face and its images under a few iterates of A,displaying the chaotic features of x ↦→ Ax. The quantization of such a linear transformation can be carried out by periodizing an... |

7 |
Itzykson C 1986 Observations sur la mécanique quantique finie
- Balian
(Show Context)
Citation Context ...ion can be carried out by periodizing any one of the standard quantization procedures in R 2 .Thishasbeen carried out first by Hannay and Berry in [16] and has since been studied by many authors; see =-=[3, 5, 12, 13, 22, 25]-=- and references therein. We will adopt the quantization procedure given by Kurlberg and Rudnick in [27]. It yields for each integer N � 1(‘N = 1/¯h’) a unitary matrix UN(A) acting on L2 (Z/NZ). Aswere... |

7 |
Itzykson C., Observations sur la mécanique quantique finie
- Balian
- 1986
(Show Context)
Citation Context |

6 |
Giannoni M J and Schmit C 1984 Characterization of chaotic quantum spectra and universality of level fluctuation laws Phys
- Bohigas
(Show Context)
Citation Context ...ystems with few degrees of freedom with chaotic classical dynamics; in fact, RMT lies at the heart of one of the basic conjectures in quantum chaos. Formulated by Bohigas, Giannoni and Schmit in 1984 =-=[8]-=-, it asserts that the eigenvalues of a quantized chaotic Hamiltonian (after suitable unfolding) behave like the spectrum of a typical member of the appropriate ensemble of random matrices. This conjec... |

6 |
Expanding graphs and invariant means, Combinatorica 17
- Shalom
- 1997
(Show Context)
Citation Context ...cate that a ‘generic’ element z in the group ring of SL(2, Z) has a spectral gap. In [14] it is proved that z has a spectral gap if the Hausdorff dimension of the limit set of Ɣz is large enough; see =-=[45]-=- for related results and [33, 34] for the discussion of this problem. We consider the quantizations Uq(z), � � � � −1 −1 Uq(z) = Uq(A1) + Uq A1 + ···+ Uq(Ak) + Uq Ak . For technical reasons, detailed ... |

6 |
Pseudo-symmetries of Anosov maps and spectral statistics, Nonlinearity 13
- Keating, Mezzadri
- 2000
(Show Context)
Citation Context ...edicted random matrix distribution for modied cat maps have been proposed. One approach,srst considered by Basilio de Matos and Ozorio de Almeido in [4], and more recently by Keating and Mezzadri in [=-=21]-=-, is to perturb a cat map by nonlinear shears; another, considered by Keppeler, Marklof, and Mezzadri in [23], is to couple a cat map with a two-spinor processing in a magneticseld. In this paper we s... |

5 |
On the quantisation of Arnold’s cat
- Knabe
- 1990
(Show Context)
Citation Context ...ion can be carried out by periodizing any one of the standard quantization procedures in R 2 .Thishasbeen carried out first by Hannay and Berry in [16] and has since been studied by many authors; see =-=[3, 5, 12, 13, 22, 25]-=- and references therein. We will adopt the quantization procedure given by Kurlberg and Rudnick in [27]. It yields for each integer N � 1(‘N = 1/¯h’) a unitary matrix UN(A) acting on L2 (Z/NZ). Aswere... |

5 |
Expanding graphs and invariant means
- Shalom
- 1996
(Show Context)
Citation Context ...dicate that a \generic" element z in the group ring of SL(2; Z) has a spectral gap. In [14] it is proved that z has a spectral gap if the Hausdor dimension of the limit set of z is large enough; =-=see [44]-=- for related results and [33, 34] for the discussion of this problem. 1 The classical limit can be thought of as a random walk supported on the toral automorphisms in question, or, following Arnold an... |

3 |
P and Mezzadri F 2000 Pseudo-symmetries of Anosov maps and spectral statistics Nonlinearity 13 747–75
- Keating
(Show Context)
Citation Context ...cted random matrix distribution for modified cat maps have been proposed. One approach, first considered by Basilio de Matos and Ozorio de Almeido in [4], and more recently by Keating and Mezzadri in =-=[21]-=-, is to perturb a cat map by nonlinear shears; another, considered by Keppeler, Marklof and Mezzadri in [23], is to couple a cat map with a two-spinor processing in a magnetic field. In this paper we ... |

3 |
Symmetric random walks on groups Trans
- Kesten
- 1959
(Show Context)
Citation Context ...similar expression for U − q (z): µ − (q−1)/4 4 � q (z) = δλj (U q − 1 − q (z)). (4) j=1 It is not difficult to show [39, 42, 15] that these converge to the measure νk(t),firstconsidered by Kesten in =-=[24]-=-, which is supported in the interval [−2 √ 2k − 1, 2 √ 2k − 1] and given by � 2k − 1 − t2 /4 dνk(t) = 2πk(1 − (t/2k) 2 dt. (5) ) Our numerical experiments, described in section 4, indicate that the un... |

3 |
Quantum boundary conditions for torus maps
- Keating, Mezzadri, et al.
- 1999
(Show Context)
Citation Context |

3 |
Quantum cat maps with spin 1=2. Nonlinearity
- Keppeler, Marklof, et al.
- 2001
(Show Context)
Citation Context ...asilio de Matos and Ozorio de Almeido in [4], and more recently by Keating and Mezzadri in [21], is to perturb a cat map by nonlinear shears; another, considered by Keppeler, Marklof, and Mezzadri in =-=[23]-=-, is to couple a cat map with a two-spinor processing in a magneticseld. In this paper we show how to recover the RMT predictions while staying within the framework of linear maps and representation t... |

2 |
Degli Esposti M and Giachetti R 1996 Quantization of a class of piecewise affine transformations on the torus Commun
- Bièvre
(Show Context)
Citation Context |