Abstract

Abstract
We describe a construction of a moduli space of solenoid functions for the C 1 +-conjugacy classes of hyperbolic dynamical systems f on surfaces with hyperbolic basic sets ?f. We explain that if the holonomies are sufficiently smooth then the diffeomorphism f is rigid in the sense that it is C 1 +conjugate to a hyperbolic affine model. We present a moduli space of measure solenoid functions for all Lipschitz conjugacy classes of C 1 +- hyperbolic dynamical systems f which have a invariant measure that is absolutely continuous with respect to Hausdorff measure. We extend Livšic and Sinai’s eigenvalue formula for Anosov diffeomorphisms which preserve an absolutely continuous measure to hyperbolic basic sets on surfaces which possess an invariant measure absolutely continuous with respect to Hausdorff measure. Introduction We say that (f, ?) is a C 1 +hyperbolic diffeomorphism if it has the following properties: (i) f: M ? M is a C 1 + adiffeomorphism of a compact surface M with respect to a C 1 + astructure on M, for some a > 0. (ii) ? is a hyperbolic invariant subset of M such that f|? is topologically transitive and ? has a local product structure. We allow both the case where ? = M and the case where ? is a proper subset of M. If ? = M then f is Anosov and M is a torus. Examples where ? is a proper subset of M include the Smale horseshoes and the codimension one attractors such as the Plykin and derived-Anosov attractors. © Mathematical Sciences Research Institute 2007.

Abstract
For a spatial stochastic epidemic model we investigate in the pair approximation scheme the differential equations for the moments. The basic reinfection model of susceptible-infected-recovered-reinfected or SIRI type is analysed, its phase transition lines calculated analytically in this pair approximation.

Abstract
We prove that the stable holonomies of a proper codimension 1 attractor Lambda, for a C-r diffeomorphism f of a surface, are not C1+theta for theta greater than the Hausdorff dimension of the stable leaves of f intersected with Lambda. To prove this result we show that there are no diffeomorphisms of surfaces, with a proper codimension 1 attractor, that are affine on a neighbourhood of the attractor and have affine stable holonomies on the attractor.

Abstract
The Von Mises-Fisher distribution defined on the hypersphere is one of the most used distributions for modelling vectorial data. If we reject the uniform distribution as a model for vectorial data, then the Von Mises-Fisher distribution is an alternative model. So we use this distribution as an alternative hypothesis in the study of the power of some tests of uniformity for some dimensions of the sphere.

Abstract
We derive likelihood ratio tests for the equality of the directional parameters of k bipolar Watson distributions defined on the hypersphere with common concentration parameter. We analyze the power of these tests in the case of two distributions supposing in the alternative hypothesis two directional parameters forming an angle, which varies from 18 degrees to 90 degrees. We also compare the likelihood ratio tests with a high-concentration F-test.