Upper bounds on the rate of quantum ergodicity

We study the semiclassical behaviour of eigenfunctions of quantum systems with ergodic classical limit. By the quantum ergodicity theorem almost all of these eigenfunctions become equidistributed in a weak sense. We give a simple derivation of an upper bound of order \abs{\ln\hbar}^{-1} on the rate of quantum ergodicity if the classical system is ergodic with a certain rate. In addition we obtain a similar bound on transition amplitudes if the classical system is weak mixing. Both results generalise previous ones by Zelditch. We then extend the results to some classes of quantised maps on the torus and obtain a logarithmic rate for perturbed cat-maps and a sharp algebraic rate for parabolic maps.Comment: 18 page

Dynamics of a higher dimensional analog of the trigonometric functions

We introduce a higher dimensional quasiregular map analogous to the trigonometric functions and we use the dynamics of this map to define, for d>1, a partition of d-dimensional Euclidean space into curves tending to infinity such that two curves may intersect only in their endpoints and such that the union of the curves without their endpoints has Hausdorff dimension one.Comment: 12 page

Analytic cliffordian functions

In classical function theory, a function is holomorphic if and only if it is complex analytic. For higher dimensional spaces it is natural to work in the context of Clifford algebras. The structures of these algebras depend on the parity of the dimension n of the underlying vector space. The theory of holomorphic Cliffordian functions reflects this dependence. In the case of odd n the space of functions is defined by an operator (the Cauchy-Riemann equation) but not in the case of even $n$. For all dimensions the powers of identity (z^n, x^n) are the foundation of function theory

On the mean square of the zeta-function and the divisor problem

Let $\Delta(x)$ denote the error term in the Dirichlet divisor problem, and $E(T)$ the error term in the asymptotic formula for the mean square of $|\zeta(1/2+it)|$. If $E^*(t) = E(t) - 2\pi\Delta^*(t/2\pi)$ with $\Delta^*(x) = -\Delta(x) + 2\Delta(2x) - {1\over2}\Delta(4x)$, then we obtain the asymptotic formula $$\int_0^T (E^*(t))^2 {\rm d} t = T^{4/3}P_3(\log T) + O_\epsilon(T^{7/6+\epsilon}),$$ where $P_3$ is a polynomial of degree three in $\log T$ with positive leading coefficient. The exponent 7/6 in the error term is the limit of the method.Comment: 10 page

On some elliptic transmission problems

Abstract. Boundary value problems for second order linear elliptic equations with coefficients having discontinuities of the first kind on an infinite number of smooth surfaces are studied. Existence, uniqueness and regularity results are furnished for the diffraction problem in such a bounded domain, and for the corresponding transmission problem in all of R N . The transmission problem corresponding to the scattering of acoustic plane waves by an infinitely stratified scatterer, consisting of layers with physically different materials, is also studied. 0. Introduction. In this work we study boundary value problems for linear equations of elliptic type whose coefficients have discontinuities of the first kind on an infinite number of smooth surfaces that divide a bounded domain in R N into nested layers. On those surfaces, the so-called "transmission (conjugacy, matching, linking) conditions" are imposed, that express the continuity of the medium and the equilibrium of the forces acting on it. The discontinuity of the coefficients of the equations corresponds to the fact that the medium consists of several physically different materials. From the point of view of the theory of generalized solutions-which we employ in our approach-such problems can be considered as special cases of usual boundary value problems. On the contrary, the investigation of these problems by classical methods requires the theory of integral equations, and in this context they differ essentially from the usual boundary value problems where the medium has smoothly varying characteristics. Boundary value problems with discontinuous coefficients (also known as diffraction problems) have been treated by many authors, employing a variety of approaches. I

Gradients and canonical transformations

Abstract. The main aim of this paper is to give some counterexamples to global invertibility of local diffeomorphisms which are interesting in mechanics. The first is a locally strictly convex function whose gradient is non-injective. The interest in this function is related to the Legendre transform. Then I show two non-injective canonical local diffeomorphisms which are rational: the first is very simple and related to the complex cube, the second is defined on the whole R 4 and is obtained from a recent important example by Pinchuk. Finally, a canonical transformation which is also a gradient (of a convex function) is provided

Uniform stability and semi-stability of motions in dynamical systems on metric spaces

Abstract. Some stability properties of motions in pseudo-dynamical systems and semi-systems are studied. Introduction. The purpose of the present paper is to study some properties of motions in dynamical and pseudo-dynamical systems (see definitions below) on metric spaces. We consider uniform stability and semistability (see Section 2) of motions in nonempty subsets of the phase space and discuss in particular properties of mappings of the types

How to get rid of one of the weights in a two-weight Poincaré inequality?

Abstract. We prove that if a Poincaré inequality with two different weights holds on every ball, then a Poincaré inequality with the same weight on both sides holds as well

On the local meromorphic extension of CR meromorphic mappings

Since the works of Trépreau, Tumanov and Jöricke, extendability properties of CR functions on a smooth CR manifold M became fairly well understood. In a natural way, 1) M is seen to be a disjoint union of CR bricks, called CR orbits, each of which being an immersed CR submanifold of M with the same CR dimension as M Key words and phrases: CR generic currents, scarred CR manifolds, removable singularities for CR functions, deformations of analytic discs, CR meromorphic mappings. We would like to mention that these removability results were originally impulsed by Jöricke in The goal of this article is to push forward meromorphic extension on CR manifolds of arbitrary codimension, the analogs of domains being wedges over CR manifolds. It seems natural to use the theory of Trépreau-Tumanov in this context. Knowing thinness of Σ f (Sarkis) and using wedge removable singularities theorems ([15], Acknowledgements. We are grateful to Professor Henkin who raised the question. We also wish to address special thanks to Frederic Sarkis. He has communicated to us the reduction of meromorphic extension of CR meromorphic mappings to a removable singularity property and we had several interesting conversations with him