arXiv subjects

Jens Marklof

Publications and source records attributed to Jens Marklof.

67 records · Page 4Linked to original sources

arXiv2006

Distribution modulo one and Ratner's theorem

Measure rigidity is a branch of ergodic theory that has recently contributed to the solution of some fundamental problems in number theory and mathematical physics. Examples are proofs of quantitative versions of the Oppenheim conjecture, related questions on the spacings between the values of quadratic forms, a proof of quantum unique ergodicity for certain classes of hyperbolic surfaces, and an approach to the Littlewood conjecture on the nonexistence of multiplicatively badly approximable numbers. In these lectures we discuss a few simple applications of one of the central results in measure rigidity: Ratner's theorem. We shall investigate the statistical properties of certain number theoretic sequences, specifically the fractional parts of $mα$, $m=1,2,3,...$, (a classical, well understood problem) and of $\sqrt{mα}$ (as recently studied by Elkies and McMullen). By exploiting equidistribution results on a certain homogeneous space, we will show that the statistical properties of these sequences can exhibit significant deviations from those of independent random variables. The ``randomness'' of other, more generic sequences such as $m^2α$ and $2^m α$ mod 1 has been studied extensively. These notes are based on lectures presented at the Institute Henri Poincare Paris, June 2005, and at the summer school `Equidistribution in number theory', CRM Montreal, July 2005. The author gratefully acknowledges support by an EPSRC Advanced Research Fellowship.

math.NT↗

arXiv2006

Explicit invariant measures for products of random matrices

We construct explicit invariant measures for a family of infinite products of random, independent, identically-distributed elements of SL(2,C). The matrices in the product are such that one entry is gamma-distributed along a ray in the complex plane. When the ray is the positive real axis, the products are those associated with a continued fraction studied by Letac and Seshadri [Z. Wahr. Verw. Geb. 62 (1983) 485-489], who showed that the distribution of the continued fraction is a generalised inverse Gaussian. We extend this result by finding the distribution for an arbitrary ray in the complex right-half plane, and thus compute the corresponding Lyapunov exponent explicitly. When the ray lies on the imaginary axis, the matrices in the infinite product coincide with the transfer matrices associated with a one-dimensional discrete Schroedinger operator with a random, gamma-distributed potential. Hence, the explicit knowledge of the Lyapunov exponent may be used to estimate the (exponential) rate of localisation of the eigenstates.

math-ph↗

arXiv2005

Unipotent flows on the space of branched covers of Veech surfaces

There is a natural action of SL(2,R) on the moduli space of translation surfaces, and this yields an action of the unipotent subgroup $U = {\begin{pmatrix} 1 & * 0 & 1 \end{pmatrix}}$. We classify the U-invariant ergodic measures on certain special submanifolds of the moduli space. (Each submanifold is the SL(2,R)-orbit of the set of branched covers of a fixed Veech surface.) For the U-action on these submanifolds, this is an analogue of Ratner's Theorem on unipotent flows. The result yields an asymptotic estimate of the number of periodic trajectories for billiards in a certain family of non-Veech rational triangles, namely, the isosceles triangles in which exactly one angle is $2 π/n$, with $n \ge 5$ and $n$ odd.

math.DS↗

arXiv2005

Quantum Leaks

We show that eigenfunctions of the Laplacian on certain non-compact domains with finite area may localize at infinity--provided there is no extreme level clustering--and thus rule out quantum unique ergodicity for such systems. The construction is elementary and based on `bouncing ball' quasimodes whose discrepancy is proved to be significantly smaller than the mean level spacing.

math-ph↗

arXiv2004

Weyl's law and quantum ergodicity for maps with divided phase space

For a general class of unitary quantum maps, whose underlying classical phase space is divided into several invariant domains of positive measure, we establish analogues of Weyl's law for the distribution of eigenphases. If the map has one ergodic component, and is periodic on the remaining domains, we prove the Schnirelman-Zelditch-Colin de Verdiere Theorem on the equidistribution of eigenfunctions with respect to the ergodic component of the classical map (quantum ergodicity). We apply our main theorems to quantised linked twist maps on the torus. In the Appendix, S. Zelditch connects these studies to some earlier results on `pimpled spheres' in the setting of Riemannian manifolds. The common feature is a divided phase space with a periodic component.

nlin.CD↗

arXiv2004

Melvin Models and Diophantine Approximation

Melvin models with irrational twist parameter provide an interesting example of conformal field theories with non-compact target space, and localized states which are arbitrarily close to being delocalized. We study the torus partition sum of these models, focusing on the properties of the regularized dimension of the space of localized states. We show that its behavior is related to interesting arithmetic properties of the twist parameter $γ$, such as the Lyapunov exponent. Moreover, for $γ$ in a set of measure one the regularized dimension is in fact not a well-defined number but must be considered as a random variable in a probability distribution.

hep-th↗

arXiv2004

Pair correlation densities of inhomogeneous quadratic forms

Under explicit diophantine conditions on $(α,β)\in\RR^2$, we prove that the local two-point correlations of the sequence given by the values $(m-α)^2+\break (n-β)^2$, with $(m,n)\in\ZZ^2$, are those of a Poisson process. This partly confirms a conjecture of Berry and Tabor [2] on spectral statistics of quantized integrable systems, and also establishes a particular case of the quantitative version of the Oppenheim conjecture for inhomogeneous quadratic forms of signature (2,2). The proof uses theta sums and Ratner's classification of measures invariant under unipotent flows.

math.NT↗

arXiv2004

Intermediate statistics in quantum maps

We present a one-parameter family of quantum maps whose spectral statistics are of the same intermediate type as observed in polygonal quantum billiards. Our central result is the evaluation of the spectral two-point correlation form factor at small argument, which in turn yields the asymptotic level compressibility for macroscopic correlation lengths.

nlin.CD↗

arXiv2003

Holomorphic almost modular forms

Holomorphic almost modular forms are holomorphic functions of the complex upper half plane which can be approximated arbitrarily well (in a suitable sense) by modular forms of congruence subgroups of large index in $\SL(2,\ZZ)$. It is proved that such functions have a rotation-invariant limit distribution when the argument approaches the real axis. An example for a holomorphic almost modular form is the logarithm of $\prod_{n=1}^\infty (1-\exp(2\piın^2 z))$. The paper is motivated by the author's studies [J. Marklof, Int. Math. Res. Not. {\bf 39} (2003) 2131-2151] on the connection between almost modular functions and the distribution of the sequence $n^2x$ modulo one.

math.NT↗

arXiv2002

Equidistribution of Kronecker sequences along closed horocycles

It is well known that (i) for every irrational number $α$ the Kronecker sequence $mα$ ($m=1,...,M$) is equidistributed modulo one in the limit $M\to\infty$, and (ii) closed horocycles of length $\ell$ become equidistributed in the unit tangent bundle $T_1 M$ of a hyperbolic surface $M$ of finite area, as $\ell\to\infty$. In the present paper both equidistribution problems are studied simultaneously: we prove that for any constant $ν> 0$ the Kronecker sequence embedded in $T_1 M$ along a long closed horocycle becomes equidistributed in $T_1 M$ for almost all $α$, provided that $\ell = M^ν \to \infty$. This equidistribution result holds in fact under explicit diophantine conditions on $α$ (e.g., for $α=\sqrt 2$) provided that $ν<1$, or $ν<2$ with additional assumptions on the Fourier coefficients of certain automorphic forms. Finally, we show that for $ν=2$, our equidistribution theorem implies a recent result of Rudnick and Sarnak on the uniformity of the pair correlation density of the sequence $n^2 α$ modulo one.

math.NT↗

arXiv2002

Pair correlation densities of inhomogeneous quadratic forms II

Denote by $\| \cdot \|$ the euclidean norm in $\RR^k$. We prove that the local pair correlation density of the sequence $\| \vecm -\vecalf \|^k$, $\vecm\in\ZZ^k$, is that of a Poisson process, under diophantine conditions on the fixed vector $\vecalf\in\RR^k$: in dimension two, vectors $\vecalf$ of any diophantine type are admissible; in higher dimensions ($k>2$), Poisson statistics are only observed for diophantine vectors of type $κ<(k-1)/(k-2)$. Our findings support a conjecture of Berry and Tabor on the Poisson nature of spectral correlations in quantized integrable systems.

math.NT↗

arXiv2000

Quantum cat maps with spin 1/2

We derive a semiclassical trace formula for quantized chaotic transformations of the torus coupled to a two-spinor precessing in a magnetic field. The trace formula is applied to semiclassical correlation densities of the quantum map, which, according to the conjecture of Bohigas, Giannoni and Schmit, are expected to converge to those of the circular symplectic ensemble (CSE) of random matrices. In particular, we show that the diagonal approximation of the spectral form factor for small arguments agrees with the CSE prediction. The results are confirmed by numerical investigations.

nlin.CD↗

arXiv1999

Quantum unique ergodicity for parabolic maps

We study the ergodic properties of quantized ergodic maps of the torus. It is known that these satisfy quantum ergodicity: For almost all eigenstates, the expectation values of quantum observables converge to the classical phase-space average with respect to Liouville measure of the corresponding classical observable. The possible existence of any exceptional subsequences of eigenstates is an important issue, which until now was unresolved in any example. The absence of exceptional subsequences is referred to as quantum unique ergodicity (QUE). We present the first examples of maps which satisfy QUE: Irrational skew translations of the two-torus, the parabolic analogues of Arnold's cat maps. These maps are classically uniquely ergodic and not mixing. A crucial step is to find a quantization recipe which respects the quantum-classical correspondence principle. In addition to proving QUE for these maps, we also give results on the rate of convergence to the phase-space average. We give upper bounds which we show are optimal. We construct special examples of these maps for which the rate of convergence is arbitrarily slow.

math-ph↗