On density of horospheres in dynamical laminations

In 1985 D.Sullivan had introduced a dictionary between two domains of complex dynamics: iterations of rational functions on the Riemann sphere and Kleinian groups. The latters are discrete subgroups of the group of conformal automorphisms of the Riemann sphere. This dictionary motivated many remarkable results in both domains, starting from the famous Sullivan's no wandering domain theorem in the theory of iterations of rational functions. One of the principal objects used in the study of Kleinian groups is the hyperbolic 3- manifold associated to a Kleinian group, which is the quotient of its lifted action to the hyperbolic 3- space. M.Lyubich and Y.Minsky have suggested to extend Sullivan's dictionary by providing an analogous construction for iterations of rational functions. Namely, they have constructed a lamination by three-dimensional manifolds equipped with a continuous family with hyperbolic metrics on them (may be with singularities). The action of the rational mapping on the sphere lifts naturally up to homeomorphic action on the hyperbolic lamination that is isometric along the leaves. The action on the hyperbolic lamination admits a well-defined quotient called {\it the quotient hyperbolic lamination}. We study the arrangement of the horospheres in the quotient hyperbolic lamination. The main result says that if a rational function does not belong to a small list of exceptions (powers, Chebyshev and Latt\`es), then there are many dense horospheres, i.e., the horospheric lamination is topologically-transitive. We show that for "many" rational functions (hyperbolic or critically-nonrecurrent nonparabolic) the quotient horospheric lamination is minimal: each horosphere is dense.Comment: The complete versio

Explicit Hermite-type eigenvectors of the discrete Fourier transform

The search for a canonical set of eigenvectors of the discrete Fourier transform has been ongoing for more than three decades. The goal is to find an orthogonal basis of eigenvectors which would approximate Hermite functions -- the eigenfunctions of the continuous Fourier transform. This eigenbasis should also have some degree of analytical tractability and should allow for efficient numerical computations. In this paper we provide a partial solution to these problems. First, we construct an explicit basis of (non-orthogonal) eigenvectors of the discrete Fourier transform, thus extending the results of [7]. Applying the Gramm-Schmidt orthogonalization procedure we obtain an orthogonal eigenbasis of the discrete Fourier transform. We prove that the first eight eigenvectors converge to the corresponding Hermite functions, and we conjecture that this convergence result remains true for all eigenvectors.Comment: 21 pages, 4 figures, 1 tabl