Proceedings of the London Mathematical Society - current issue

Deformations of finite conformal energy: existence and removability of singularities

Iwaniec, T., Onninen, J. — Wed, 30 Dec 2009 06:22:52 PST

This paper features a class of mappings between bounded domains X, Y Rⁿ, having finite n-harmonic energy, such that we have The fundamental question is whether or not the domains X, Y Rⁿ of the same topological type admit a homeomorphism in a given homotopy class having finite energy. The examples of non-existence, somewhat testing our theory, arise when we remove from bounded smooth domains X and Y thin subsets and , referred to as cracks or fractures. We are looking for homeomorphisms of finite energy for which is the cluster set of h over . In general, infinite energy is required in order to increase the dimension of a crack that is, when . Suppose now that a bounded deformation of finite energy is given. Does h extend continuously to X and, if so, is the extension injective on X? We give affirmative answers to these questions.

Families of absolutely simple hyperelliptic jacobians

Zarhin, Y. G. — Wed, 30 Dec 2009 06:22:52 PST

We prove that the jacobian of a hyperelliptic curve y² = (x – t)h(x) has no non-trivial endomorphisms over an algebraic closure of the ground field K of characteristic zero if t K and the Galois group of the polynomial h(x) over K is an alternating or symmetric group on deg(h) letters and deg(h) is an even number greater than 8. (The case of odd deg(h) > 3 follows easily from previous results of the author.)

Spectral properties of matrices associated with some directed graphs

Davies, E. B., Incani, P. A. — Wed, 30 Dec 2009 06:22:52 PST

We study the spectral properties of certain non-self-adjoint matrices associated with large directed graphs. Asymptotically the eigenvalues converge to certain curves, apart from a finite number that have limits not on these curves.

A Maslov cocycle for unitary groups

Kramer, L., Tent, K. — Wed, 30 Dec 2009 06:22:52 PST

We introduce a 2-cocycle for symplectic and skew-hermitian hyperbolic groups over arbitrary fields and skew-fields, with values in the Witt group of hermitian forms. This cocycle has good functorial properties: it is natural under extension of scalars and stable, and so it can be viewed as a universal 2-dimensional characteristic class for these groups. Over R and C, it coincides with the first Chern class.

Theorie ergodique des fractions rationnelles sur un corps ultrametrique

Favre, C., Rivera-Letelier, J. — Wed, 30 Dec 2009 06:22:52 PST

Résumé

On donne les premiers éléments pour l’étude des propriétés ergodiques d’une fraction rationnelle à coefficients dans un corps algébriquement clos et complet pour une norme non archimédienne. En particulier, pour une telle fraction rationnelle R on montre l’existence d’une mesure naturelle _R représentant la distribution asymptotique des préimages itérées de chaque point non exceptionnel de R. On montre que cette mesure est (exponentiellement) mélangeante, et qu’elle satisfait au théorème limite central. De plus, on donne une estimation de l’entropie métrique de cette mesure, et de l’entropie topologique de R, qui permettent de caractériser les fractions rationnelles d’entropie topologique nulle.

We make the first steps towards an understanding of the ergodic properties of a rational map defined over a complete algebraically closed non-archimedean field. For such a rational map R, we construct a natural invariant probability measure _R which represents the asymptotic distribution of preimages of non-exceptional points. We show that this measure is exponentially mixing, and satisfies the central limit theorem. We prove some general bounds on the metric entropy of _R, and on the topological entropy of R. We finally prove that rational maps with vanishing topological entropy have potential good reduction.

The true complexity of a system of linear equations

Gowers, W. T., Wolf, J. — Wed, 30 Dec 2009 06:22:52 PST

In this paper we look for conditions that are sufficient to guarantee that a subset A of a finite Abelian group G contains the ‘expected’ number of linear configurations of a given type. The simplest non-trivial result of this kind is the well-known fact that if G has odd order, A has density and all Fourier coefficients of the characteristic function of A are significantly smaller than (except the one at zero, which equals ), then A contains approximately ³|G|² triples of the form (a, a+d, a+2d). This is ‘expected’ in the sense that a random set A of density has approximately ³|G|² such triples with very high probability. More generally, it was shown by the first author (in the case G = Z_N for N prime, but the proof generalizes) that a set A of density has about ^k|G|² arithmetic progressions of length k if the characteristic function of A is almost as small as it can be, given its density, in a norm that is now called the U^k–1-norm. When investigating linear equations in the primes, Green and Tao found the most general statement that follows from the technique used to prove this result, introducing a notion that they call the complexity of a system of linear forms. They prove that if A has almost minimal U^k+1-norm, then it has the expected number of linear configurations of a given type, provided that the associated complexity is at most k. The main result of this paper is that the converse is not true: in particular there are certain systems of complexity 2 that are controlled by the U²-norm, whereas the result of Green and Tao requires the stronger hypothesis of U³-control. We say that a system of m linear forms L₁, ..., L_m in d variables with integer coeffcients has true complexity k if k is the smallest positive integer such that, for any set A of density and almost minimal U^k+1-norm, the number of d-tuples (x₁, ..., x_d) such that L_i(x₁, ..., x_d) A for every i is approximately ^m|G|^d. We conjecture that the true complexity k is the smallest positive integer s for which the functions L₁^s+1, ... ,L_m^s+1 are linearly independent. Using the ‘quadratic Fourier analysis’ of Green and Tao we prove this conjecture in the case where the complexity of the system (in Green and Tao's sense) is 2, s=1 and G is the group F_pⁿ for some fixed odd prime p. A closely related result in ergodic theory was recently proved independently by Leibman. We end the paper by discussing the connections between his result and ours.

Peano's theorem for rough differential equations in infinite-dimensional Banach spaces

Caruana, M. — Wed, 30 Dec 2009 06:22:52 PST

We present a proof for Peano's theorem for rough differential equations, which is valid in infinite dimensions under an appropriate compactness assumption on the vector fields. Our approach makes full use of Lyons’ Universal Limit Theorem and is based on the construction of a family of rough polynomial approximations, each of which is a concatenation of rough path solutions of different equations.

Primitive permutation groups of bounded orbital diameter

Liebeck, M. W., Macpherson, D., Tent, K. — Wed, 30 Dec 2009 06:22:52 PST

We give a description of infinite families of finite primitive permutation groups for which there is a uniform finite upper bound on the diameter of all orbital graphs. This is equivalent to describing families of finite permutation groups such that every ultraproduct of the family is primitive. A key result is that, in the almost simple case with socle of fixed Lie rank, apart from very specific cases, there is such a diameter bound. This is proved using recent results on the model theory of pseudofinite fields and difference fields.

Rigidity of measures invariant under semisimple groups in positive characteristic

Einsiedler, M., Ghosh, A. — Wed, 30 Dec 2009 06:22:52 PST

M. Ratner has conjectured a positive characteristic version of her seminal results classifying orbit closures and invariant measures of unipotent flows on homogeneous spaces. In this paper, we provide a partial answer by establishing a positive characteristic version of her classification result for measures invariant under semisimple groups.

Energy measures and indices of Dirichlet forms, with applications to derivatives on some fractals

Hino, M. — Wed, 30 Dec 2009 06:22:52 PST

We introduce the concept of index for regular Dirichlet forms by means of energy measures, and discuss its properties. In particular, it is proved that the index of strong local regular Dirichlet forms is identical with the martingale dimension of the associated diffusion processes. As an application, a class of self-similar fractals is taken up as an underlying space. We prove that first-order derivatives can be defined for functions in the domain of the Dirichlet forms and their total energies are represented as the square integrals of the derivatives.