New progress in the theory of homogeneous flows

Bäcklund transformations of solutions of nonlinear evolution equations and the Lie Bianchi transformation

Introduction During the last 15–20 years it has been realized that certain problems in Diophantine approximation and number theory can be solved using geometry of the space of lattices and methods from the theory of flows on homogeneous spaces. The purpose of this survey is to demonstrate this approach on several examples. We will start with Diophantine approximation on manifolds where we will briefly describe the proof of Baker–Sprindžuk conjectures and some Khintchine-type theorems. The next topic is the Oppenheim conjecture proved in the mid-1980s and the Littlewood conjecture, still not settled. After that we will go to quantitative generalizations of the Oppenheim conjecture and to counting lattice points on homogeneous varieties. In the last part we will discuss results on unipotent flows on homogeneous spaces which play, directly or indirectly, the most essential role in the solution of the above-mentioned problems. Most of those results on unipotent flows are proved using ergodic theorems and also notions such as minimal sets and invariant measures. These theorems and notions have no effective analogs and because of that the homogeneous space approach is not effective in a certain sense. We will briefly discuss the problem of the effectivization at the very end of the paper. The author would like to thank A. Eskin and D. Kleinbock for their comments on a preliminary version of this article. Diophantine approximation on manifolds We start by recalling some standard notation and terminology.

The purpose of this work is to present an algorithm to determine the motion of a single hydrodynamic vortex on a closed surface of constant curvature and of genus greater than one. The algorithm is based on a relation between the Laplace-Beltrami Green function and the heat kernel. The algorithm is used to compute the motion of a vortex on the Bolza surface. This is the first determination of the orbits of a vortex on a closed surface of genus greater than one. The numerical results show that all the 46 vortex equilibria can be explicitly computed using the symmetries of the Bolza surface. Some of these equilibria allow for the construction of the first two examples of infinite vortex crystals on the hyperbolic disc. The following theorem is proved: 'a Weierstrass point of a hyperellitic surface of constant curvature is always a vortex equilibrium'.

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 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.

A one-parameter family of time reversible Anosov flows is studied; physically, it describes a particle moving on a surface of constant negative curvature under the action of an electric field (corresponding to an automorphic form) and of a ‘thermostatting’ force (given by Gauss's least-constraint principle). We show that the flows are dissipative, in the sense that the average volume contraction rate is positive and the Sinai–Ruelle–Bowen measure is singular with respect to the volume: therefore they verify the assumptions for the validity of the continuous time version of Gallavotti–Cohen's fluctuation theorem for the large fluctuations of the average volume contraction rate. If several independent electric fields are considered, it makes sense to ask for the validity of Onsager's reciprocity: we show, by explicitly computing the relevant transport coefficents, that it is indeed obeyed.

Expansiveness for the geodesic and horocycle flows on compact Riemann surfaces of constant negative curvature

Chaotic dynamical systems can be studied using the example of a surface of constant negative curvature. Of particular interest are tori with one exceptional point, because the motion of a particle on such a surface is very close to the scattering of an electron on a small molecule. The explicit calculations require a surface which is compatible with the modular group. The construction of the known four cases is carried out with the help of elementary number theory, and the scattering function for the solutions of the Laplace operator is obtained. The geometrical discussion leads to tori with two exceptional points, as well as a special example of the latter which is compatible with the modular group and yet does not belong to a torus with only one exceptional point. A rather unusual representation of the general Fricke-Klein groups in terms of 4 by 4 matrices is also given, which is rational in two of the three traces A, B, and C, and does not use the third one.KeywordsFundamental DomainNegative CurvatureDiophantine EquationModular GroupCommutator SubgroupThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We consider flows, called Wu flows, whose orbits are the unstable manifolds of a codimension one Anosov flow. Under some regularity assumptions, we give a short proof of the strong mixing property of Wuflows and we show that Wu flows have purely absolutely continuous spectrum in the orthocomplement of the constant functions. As an application, we obtain that time changes of the classical horocycle flows for compact surfaces of constant negative curvature have purely absolutely continuous spectrum in the orthocomplement of the constant functions for time changes in a regularity class slightly less than C2. This generalises recent results on time changes ofhorocycle 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.

There is a natural action of SL $(2,\mathbb{R})$ on the moduli space of translation surfaces, and this yields an action of the unipotent subgroup $U = \big\{\big(\begin{smallmatrix}1 & * 0 & 1\end{smallmatrix}\big)\big\}$ . We classify the U -invariant ergodic measures on certain special submanifolds of the moduli space. (Each submanifold is the SL $(2,\mathbb{R})$ -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 $2 \pi/n$ , with $n \ge 5$ and n odd.

Chapter 13 Pointwise ergodic theorems for actions of groups

We study a refined version of Linnik's problem on the asymptotic behavior of the number of representations of integers m by an integral polynomial as m tends to infinity. Assuming that the polynomials arise from invariant theory, we reduce the question to the study of limiting behavior of measures invariant under unipotent flows. Our main tool is then Ratner's theorem on the uniform distribution of unipotent flows, in a form refined by Dani and Margulis [DM2]

In this paper we investigate the relationship between two topics which at first sight seem unrelated. The first deals with ergodic properties of geodesic flows on two-dimensional surfaces of constant negative curvature, a rather active area in the thirties studied by many well known mathematicians. For a detailed survey of the work during that period see [H2]. The second one deals with ergodic properties of noninvertible mappings of the unit interval, a current popular subject and one also with an interesting history going back to Gauss (See [B]). We shall show how each of these subjects sheds light on the other. Ergodic properties of interval maps can be used to prove ergodicity of the flows and conversely. Furthermore, explicit formulas for invariant measures of interval maps can be derived from the invariance of hyperbolic measure for the flows. (Actually we could go a step further and trace a connection of these formulas to Liouville’s theorem for Hamiltonian Systems.) This fact is particularly interesting as there is a paucity of explicit formulas for invariant measures of interval maps and we have here a method of deriving a class of these. In particular we shall show how Gauss’s formula for the invariant measure associated with continued fractions, which seems to have been produced ad hoc, can be derived anew.

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.