The classical r-matrix method for nonlinear sigma-model

The canonical Poisson structure of nonlinear sigma-model is presented as a Lie-Poisson r-matrix bracket on coadjoint orbits. It is shown that the Poisson structure of this model is determined by some `hidden singularities' of the Lax matrix.Comment: 18 pages, LaTe

Symmetric measures via moments

Algebraic tools in statistics have recently been receiving special attention and a number of interactions between algebraic geometry and computational statistics have been rapidly developing. This paper presents another such connection, namely, one between probabilistic models invariant under a finite group of (non-singular) linear transformations and polynomials invariant under the same group. Two specific aspects of the connection are discussed: generalization of the (uniqueness part of the multivariate) problem of moments and log-linear, or toric, modeling by expansion of invariant terms. A distribution of minuscule subimages extracted from a large database of natural images is analyzed to illustrate the above concepts.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ6144 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

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