Surfaces Of Constant Negative Curvature Research Articles

Turbulence on Hyperbolic Plane: The Fate of Inverse Cascade

We describe ideal incompressible hydrodynamics on the hyperbolic plane which is an infinite surface of constant negative curvature. We derive equations of motion, general symmetries and conservation laws, and then consider turbulence with the energy density linearly increasing with time due to action of small-scale forcing. In a flat space, such energy growth is due to an inverse cascade, which builds a constant part of the velocity autocorrelation function proportional to time and expanding in scales, while the moments of the velocity difference saturate during a time depending on the distance. For the curved space, we analyze the long-time long-distance scaling limit, that lives in a degenerate conical geometry, and find that the energy-containing mode linearly growing with time is not constant in space. The shape of the velocity correlation function indicates that the energy builds up in vortical rings of arbitrary diameter but of width comparable to the curvature radius of the hyperbolic plane. The energy current across scales does not increase linearly with the scale, as in a flat space, but reaches a maximum around the curvature radius. That means that the energy flux through scales decreases at larger scales so that the energy is transferred in a non-cascade way, that is the inverse cascade spills over to all larger scales where the energy pumped into the system is cumulated in the rings. The time-saturated part of the spectral density of velocity fluctuations contains a finite energy per unit area, unlike in the flat space where the time-saturated spectrum behaves as k^{-5/3}.

Limit Theorems for Horocycle Flows

The main results of this paper are limit theorems for horocycle flows on compact surfaces of constant negative curvature. One of the main objects of the paper is a special family of horocycle-invariant finitely additive Holder measures on rectifiable arcs. An asymptotic formula for ergodic integrals for horocycle flows is obtained in terms of the finitely-additive measures, and limit theorems follow as a corollary of the asymptotic formula. The objects and results of this paper are similar to those in [15], [16], [4] and [5] for translation flows on flat surfaces. The arguments are based on the classification of invariant distributions for horocycle flows established in [12].

Classical nonlinear response of a chaotic system. I. Collective resonances

We develop a general semiquantitative picture of nonlinear classical response in strongly chaotic systems. In contrast to behavior in integrable or almost integrable systems, the nonlinear classical response in chaotic systems vanishes at long times. The exponential decay of the response functions in the case of strong chaos is attributed to both exponentially decaying and growing elements in the stability matrices. We calculate the linear and second-order response in one of the simplest chaotic systems: free classical motion on a compact surface of constant negative curvature. The response reveals certain features of collective resonances which do not correspond to any periodic classical trajectories. We demonstrate the relevance of the model for the interpretation of spectroscopic experiments.

Quantization of periodic motions on compact surfaces of constant negative curvature in a magnetic field

We use the semiclassical approach to study the spectral problem for the Schrodinger operator of a charged particle confined to a two-dimensional compact surface of constant negative curvature. We classify modes of classical motion in the integrable domain E < E cr and obtain a classification of semiclassical solutions as a consequence. We construct a spectral series (spectrum part approximated by semiclassical eigenvalues) corresponding to energies not exceeding the threshold value E cr; the degeneration multiplicity is computed for each eigenvalue.

Deformation of $\Gamma_0(5)$-cusp forms

We develop an algorithm for numerical computation of the Eisenstein series on a Riemann surface of constant negative curvature. We focus in particular on the computation of the poles of the Eisenst ...

Symbolic dynamics for the modular surface and beyond

In this expository article we describe the two main methods of representing geodesics on surfaces of constant negative curvature by symbolic sequences and their development. A geometric method stems from a 1898 work of J. Hadamard and was developed by M. Morse in the 1920s. It consists of recording the successive sides of a given fundamental region cut by the geodesic and may be applied to all finitely generated Fuchsian groups. Another method, of arithmetic nature, uses continued fraction expansions of the end points of the geodesic at infinity and is even older—it comes from the Gauss reduction theory. Introduced to dynamics by E. Artin in a 1924 paper, this method was used to exhibit dense geodesics on the modular surface. For 80 years these classical works have provided inspiration for mathematicians and a testing ground for new methods in dynamics, geometry and combinatorial group theory. We present some of the ideas, results (old and recent), and interpretations that illustrate the multiple facets of the subject.

A new characterization of Gromov hiperbolicity for negatively curved surfaces

In this paper we show that to check Gromov hyperbolicity of any surface of constant negative curvature, or, Riemann surface, we only need to verify the Rips condition on a very small class of triangles, namely, those obtained by marking three points in a simple closed geodesic. This result is, in fact, a new characterization of Gromov hyperbolicity for Riemann surfaces.

Unique ergodicity, stable ergodicity, and the Mautner phenomenon for diffeomorphisms

In 1954, F. Mautner gave a simple representation theoretic argument that for compact surfaces of constant negative curvature, invariance of a function along the geodesic flow implies invariance along the horocycle flows (these are facts which imply ergodicity of the geodesic flow itself), [M]. Many generalizations of this Mautner phenomenon exist in representation theory, [St1]. Here, we establish a new generalization, Theorem 2.1, whose novelty is mostly its method of proof, namely the Anosov-Hopf ergodicity argument from dynamical systems. Using some structural properties of Lie groups, we also show that stable ergodicity is equivalent to the unique ergodicity of the strong stable manifold foliations in the context of affine diffeomorphisms.

Plasmon-polaritons on the surface of a pseudosphere

Plasmon-polariton behavior on the surfaces of constant negative curvature is considered. Analytical solutions for the eigenfrequencies and eigenfunctions are found, which describe resonant behavior of metal nanotips and nanoholes. These resonances may be used to improve performance of metal tips and nanoapertures in various scanning probe microscopy and surface-enhanced spectroscopy techniques.

Manifolds of infinite topological type with integrable geodesic flows

Let (M,k) be a complete surface of constant negative curvature (resp. an Open image in new window-geometric 3-manifold). This paper constructs a complete riemannian 8-manifold (resp. 9-manifold) (Σ,h) such that Σ is homotopy equivalent to M, the geodesic flow of h is completely integrable and there is a riemannian embedding (M,k)Open image in new window(Σ,h). This embeds the geodesic flow of (M,k) as a subsystem of an integrable geodesic flow. Amongst the manifolds Σ is an 8-dimensional manifold whose fundamental group is the free group on countably many generators.

Charge Fluctuations for a Coulomb Fluid in a Disk on a Pseudosphere

The classical (i.e., non-quantum) equilibrium statistical mechanics of a Coulomb fluid living on a pseudosphere (an infinite surface of constant negative curvature) is considered. The Coulomb fluid occupies a large disk communicating with a reservoir (grand-canonical ensemble). The total charge Qon the disk fluctuates. In a macroscopic description, the charge correlations near the boundary circle can be described as correlations of a surface charge density σ. In a macroscopic approach, the variance of Qand the correlation function of σare computed; they are universal. These macroscopic results are shown to be valid for two solvable microscopic models, in the limit when the microscopic thickness of the surface charge density goes to zero.

Nandor Balazs

Nandor Balazs

Schottky-type groups and minimal sets of horocycle and geodesic flows

In the first part of the paper the following conjecture stated by Dal'bo and Starkov is proved: the geodesic flow on a surface of constant negative curvature has a non-compact non-trivial minimal set if and only if the Fuchsian group is infinitely generated or contains a parabolic element. In the second part interesting examples of horocycle flows are constructed: 1) a flow whose restriction to the non-wandering set has no minimal subsets, and 2) a flow without minimal sets. In addition, an example of an infinitely generated discrete subgroup of with all orbits discrete and dense in is constructed.

Optimal rotations of deformable bodies and orbits in magnetic fields.

Deformations can induce rotation with zero angular momentum where dissipation is a natural "cost function." This gives rise to an optimization problem of finding the most effective rotation with zero angular momentum. For certain plastic and viscous media in two dimensions the optimal path is the orbit of a charged particle on a surface of constant negative curvature with a magnetic field whose total flux is half a quantum unit.

Leafwise cohomology and rigidity of certain Lie group actions

A locally free Lie group action on a closed manifold is called parameter rigid, if given another action with the same orbit foliation, there is a orbit-preserving diffeomorphism which conjugates one action to the other, up to an automorphism of the Lie group. We show some relationships between parameter rigidity and the first leafwise cohomology of the orbit foliation, and give a new example of a parameter rigid solvable group action. We also compute the first leafwise cohomology of the stable foliation of geodesic flows of closed orientable surfaces of constant negative curvature.

Magnetic moment of an electron gas on the surface of constant negative curvature

The magnetic moment of an electron gas on the surface of constant negative curvature is investigated. It is shown that the surface curvature leads to the appearance of the region of the monotonic dependence M(B) at low magnetic fields. At high magnetic fields, the dependence of the magnetic moment on a magnetic field is the oscillating one. The effect of the surface curvature is to increase the region of the monotonic dependence of the magnetic moment and to break the periodicity of oscillations of the magnetic moment as a function of an inverse magnetic field.

Quantum Hall effect on the Lobachevsky plane

The Hall conductivity of an electron gas on the surface of constant negative curvature (the Lobachevsky plane) in the presence of an orthogonal magnetic field is investigated. It is shown that the effect of the surface curvature is to change the break locations and the plateau widths in the Hall conductivity. An increase of temperature results in smearing of the steps.

Pressures for a One-Component Plasma on a Pseudosphere

The classical (i.e., non-quantum) equilibrium statistical mechanics of a two-dimensional one-component plasma (a system of charged point-particles embedded in a neutralizing background) living on a pseudosphere (an infinite surface of constant negative curvature) is considered. In the case of a flat space, it is known that, for a one-component plasma, there are several reasonable definitions of the pressure, and that some of them are not equivalent to each other. In the present paper, this problem is revisited in the case of a pseudosphere. General relations between the different pressures are given. At one special temperature, the model is exactly solvable in the grand canonical ensemble. The grand potential and the one-body density are calculated in a disk, and the thermodynamic limit is investigated. The general relations between the different pressures are checked on the solvable model.

Landau levels in the noncommutativeAdS2

We formulate the Landau problem in the context of the noncommutative analog of a surface of constant negative curvature, that is $AdS_2$ surface, and obtain the spectrum and contrast the same with the Landau levels one finds in the case of the commutative $AdS_2$ space.

Action correlation of orbits through non-conventional time reversal

Recently, Sieber and Richter calculated semiclassically a first off-diagonal contribution to the orthogonal form factor for a billiard on a surface of constant negative curvature. Following prior suggestions from the theory of disordered systems, they considered orbit pairs with almost the same action. For a generalization to systems invariant under non-conventional time reversal, which also belong to the orthogonal symmetry class, we show here that it is necessary to redefine the configuration space by a suitable canonical transformation; the distinction of this space is that it lets time reversal look conventional.