Surfaces Of Constant Negative Curvature Research Articles

SPIKY HIGHER GENUS STRINGS

It is clear from both the non-perturbative and perturbative approaches to two-dimensional quantum gravity that a new strong coupling regime is setting in at d = 1, independent of the genus of the worldsheet being considered. It has been suggested that a Kosterlitz–Thouless (KT) phase transition in the Liouville theory is the cause of this behavior. However, it has recently been pointed out that the XY-model, which displays a KT transition on the plane and the sphere, is always in the strong coupling, disordered phase on a surface of constant negative curvature. A higher genus worldsheet can be represented as a fundamental region on just such a surface, which might seem to suggest that the KT picture predicts a strong coupling region for arbitrary d, contradicting the known results. We resolve the apparent paradox.

An analogue of Mertens' theorem for closed orbits of Axiom A flows

For an Axiom A flow restricted to a basic set we prove an analogue of Mertens' theorem of prime number theory. The result is also established for the geodesic flow on a non-compact, finite area surface of constant negative curvature. Applying this to the modular surface yields some asymptotic formulae concerning quadratic forms.

Exact theory for the quantum eigenstates of a strongly chaotic system

We present an exact theory for the quantum eigenstates of a strongly chaotic system, which consists of a point particle sliding freely on a two-dimensional compact surface of constant negative curvature. The main result is a general sum rule which relates an (almost) arbitrarily smoothed sum over the quantum wavefunctions to a sum over the lengths of the classical orbits. As an example, we apply this formula to the Gaussian smoothed sum over wavefunctions. The question of “scars” as imprints of individual closed orbits is discussed, and the connection with Bogomolny's semiclassical theory is explicitly worked out. The investigation of the amplitude distribution P(Ψ) reveals that the higher excited wavefunctions Ψ behave as Gaussian random waves.

Bäcklund transformations for surfaces in Minkowski space

A Bäcklund transformation is constructed between spacelike surfaces of constant negative curvature and timelike surfaces of constant negative curvature in three-dimensional Minkowksi space. The transformation gives a differential geometric interpretation to a Bäcklund transformation between the elliptic sine-Gordon equation and the elliptic sinh-Gordon equation studied by Leibbrandt [J. Math. Phys. 19, (1978)].

The Differential Zeta Function for Axiom A Attractors

closed orbit of least period X( T). An important problem is to construct the meromorphic extensions for these zeta functions. A detailed knowledge of the domain of '(s) gives information on the long-term behaviour of the For example, this is a standard approach to deriving asymptotic formulae for closed orbits; cf. [9]. For the case of geodesic flows on surfaces of constant negative curvature, the zeta function is known to have a meromorphic extension to the entire complex plane. Unfortunately, an obstruction to a general theory along these lines is the existence of examples of Axiom A flows for which D(s) does not have a meromorphic extension to C. In a recent article, William Parry proposed a modified definition of a zeta function for Axiom A attractors. He defined the differential zeta function by Ns(s) = H(I e-S(T)<l where the least period X(X) of a closed orbit is replaced by the expansion Xu(T) (along the unstable manifold) around the orbit [8]. Parry also defined the useful notion of a synchronized flow (i.e., one for which the measure of maximal entropy and the SRB measure coincide). The following remark is taken from page 273 of his article: When the flow is synchronized ... the zeta functions coincide. This is the case, for example, with a geodesic flow on a surface of constant negative curvature. One wonders whether synchronization has something to do with the very desirable features of the zeta function of such a geodesic flow. In this article we shall give a solution to Parry's question which shows that in many cases DU(s) has a meromorphic extension to the entire complex plane. Our main result can be stated as follows:

Time delay, resonances, Riemann zeros and chaos in a model quantum scattering system

The quantum treatment of an intrinsically chaotic model scattering system originally studied by Gutzwiller (1983) is extended to include the time delay and to make explicit the zeros of the Riemann zeta function in the mathematical expressions for the scattering matrix S and the time delay. The system consists of a particle moving on a two-dimensional surface of constant negative curvature. The scattering in this unusual system is dominated by resonances associated with poles of S in the complex momentum plane, the real parts of these poles are one-half of the imaginary parts of the Riemann zeros. The resonances have a constant width but their average spacing varies with the momentum. The focal point is the manifestation of chaotic behaviour in the time delay. Features considered in this regard include: the momentum dependence of the time delay in regions of overlapping and isolated resonances, the decomposition of the time delay into an average (dynamical) component and a fluctuating (chaotic) component, and characterisation of the fluctuating component by its autocorrelation function.

On the periodic orbits of a strongly chaotic system

A point particle sliding freely on a two-dimensional surface of constant negative curvature (Hadamard-Gutzwiller model) exemplifies the simplest chaotic Hamiltonian system. Exploiting the close connection between hyperbolic geometry and the group SU(1,1)/⦅±1⦆, we construct an algorithm (symboliv dynamics), which generates the periodic orbits of the system. For the simplest compact Riemann surface having as its fundamental group the “octagon group”, we present an enumeration of more than 206 million periodic orbits. For the length of the nth primitive periodic orbit we find a simple expression in terms of algebraic numbers of the form m + √2 n ( m, n ϵN are governed by a particular Beatty sequence), which reveals a strange arithmetical structure of chaos. Knowledge of the length spectrum is crucial for quantization via the Selberg trace formula (periodic orbit theory), which in turn is expected to unravel the mystery of quantum chaos.

Quantum chaos of the Hadamard-Gutzwiller model.

We present a theory of quantum chaos of the Hadamard-Gutzwiller model, a quantum mechanical system which describes the motion of a particle on a surface of constant negative curvature. The theory is based on periodic-orbit sum rules that can be rigorously derived from the Selberg trace formula and which provide an exact substitute, appropriate for our strongly chaotic system, for the Bohr-Sommerfeld-Einstein quantization rules. Our recent enumeration of the classical periodic orbits enables us to evaluate the sum rules numerically and to demonstrate thereby that the theory provides also a practical method to study the quantum chaos of spectra.

Some geometrical models of chaotic dynamics

The free motion of a particle on a surface of constant negative curvature (a pseudosphere) was one of the first models of chaotic motion. It became the prototype for the theory of hyperbolic systems developed by Bowen and Sinai. In these models, geometry suggests a symbolic coding which already exhibits fully chaotic behaviour. One can return to these models to seek possible manifestations of quantum chaos. Here the mathematical technique is harmonic analysis on hyperbolic space. Chaotic behaviour seems to appear both in the behaviour of individual eigenfunctions and in the sequence of spectral values.

On Selberg's zeta function for compact Riemann surfaces

We present several novel relations for Selberg's zeta function for compact Riemann surfaces. The results find direct application in calculating determinants entering multiloop amplitudes in string theory, and in determining the energy spectrum of chaotic motion on a surface of constant negative curvature.

Spectral fluctuations and zeta functions

The study theoretically and numerically the role of the fluctuations of eigenvalue spectra /..mu../sub n/ in a particular analytical continuation process applied to the (generalized) zeta function Z(s) = ..sigma../sub n/..mu../sub n//sup -s/ for s large and positive. A particularly interesting example is the spectrum of the Laplacian on a triangular domain which tessellates a compact surface of constant negative curvature (of genus two). The authors indeed find that the fluctuations restrict the abscissa of convergence, and also affect the rate of convergence. This then initiates a new approach to the exploration of spectral fluctuations through the convergence of analytical continuation processes.

Margulis distributions for Anosov flows

For the strong unstable foliation (or horocycle foliation) of an Anosov flow there exists a unique transverse measure called the Margulis measure. In this paper we extend Margulis' results to more general “transverse distributions” for the foliation. As an application we derive our main result: The non-zero analytic extension to a strip of the Selberg zeta function for compact surfaces of constant negative curvature persists under small perturbations in the metric. There is an equivalent formulation in terms of the Fourier transform of the correlation function.

Geometrical Markov coding of geodesics on surfaces of constant negative curvature

AbstractA natural geometrical representation of the geodesic flow on a surface M of constant negative curvature is given in which the base transformation is the shift on a (finite type) space of shortest words relative to a fixed generating set for π1(M) and the height function is the hyperbolic distance across a fundamental region for π1(M). This representation is obtained by comparing cutting sequences on M with generalised continued fraction expansions of endpoints on ℝ

Lyapunov Exponents for a Stochastic Analogue of the Geodesic Flow

New invariants for a Riemannian manifold are defined as Lyapunov exponents of a stochastic analogue of the geodesic flow. A lower bound is given reminiscent of corresponding results for the geodesic flow, and an upper bound is given for surfaces of positive curvature. For surfaces of constant negative curvature a direct method via the Doob $h$-transform is used to determine the full Lyapunov structure relating the stable manifolds to the horocycles.

Asymptotic distribution of closed geodesics

In this paper the distribution of closed geodesics on surfaces of constant negative curvature are studied from a dynamical viewpoint. Asymptotic estimates are derived independently of the work of Selberg or Margulis, or the work of Bowen on Axiom A flows.

Geometrical interpretation of the solutions of the sine–Gordon equation

The sine–Gordon equation is known to possess solutions that correspond to solitons, that is, localized entities that maintain their shape after collisions, and have certain properties characteristic of elementary particles. Although the algebraic structure of these solutions is well known, their geometric interpretation as surfaces of constant negative curvature has not been previously illuminated. We discuss these surfaces herein. Curves drawn on these surfaces along the asymptotic directions at each point simulate solutions of the nonlinear wave equation φxx−φtt=sin φ.

Perturbations of geodesic flows on surfaces of constant negative curvature and their mixing properties: 486 (M1,T2) P. Collet, Laboratoire de Physique Theorique, Ecole Polytechnique, 91128 Palaiseau, FRANCE, and IMA; H. Epstein, IHES, 91440, Bures S/Yvette, FRANCE; G. Gallavotti, Istituto Matematico, Universita di Roma, ITALY.

Perturbations of geodesic flows on surfaces of constant negative curvature and their mixing properties: 486 (M1,T2) P. Collet, Laboratoire de Physique Theorique, Ecole Polytechnique, 91128 Palaiseau, FRANCE, and IMA; H. Epstein, IHES, 91440, Bures S/Yvette, FRANCE; G. Gallavotti, Istituto Matematico, Universita di Roma, ITALY.

Rational ergodicity of geodesic flows

AbstractWe prove the rational egodicity of geodesic flows on divergence type surfaces of constant negative curvature, and identify their asymptotic types.

Perturbations of geodesic flows on surface of constant negative curvature and their mixing properties

We consider one parameter analytic hamiltonian perturbations of the geodesic flows on surfaces of constant negative curvature. We find two different necessary and sufficient conditions for the canonical equivalence of the perturbed flows and the non-perturbed ones. One condition says that the “Hamilton-Jacobi equation” (introduced in this work) for the conjugation problem should admit a solution as a formal power series (not necessarily convergent) in the perturbation parameter. The alternative condition is based on the identification of a complete set of invariants for the canonical conjugation problem. The relation with the similar problems arising in the KAM theory of the perturbations of quasi periodic hamiltonian motions is briefly discussed. As a byproduct of our analysis we obtain some results on the Livscic, Guillemin, Kazhdan equation and on the Fourier series for the SL(2, ℝ) group. We also prove that the analytic functions on the phase space for the geodesic flow of unit speed have a mixing property (with respect to the geodesic flow and to the invariant volume measure) which is exponential with a universal exponent, independent on the particular function, equal to the curvature of the surface divided by 2. This result is contrasted with the slow mixing rates that the same functions show under the horocyclic flow: in this case we find that the decay rate is the inverse of the time (“up to logarithms”).

Dense-packed arrays on surfaces of constant negative curvature

Dense-packed assemblies of hard spheres are standard models of disorder in metallic glasses. Such packings are difficult to obtain with identical disks on flat two-dimensional surfaces because triangular packing units are easily incorporated into a hexagonal close-packed lattice. By packing disks on a two-dimensional manifold of constant negative curvature, however, one can study packing problems quite similar to those found in three-dimensional flat space.