Narrow Results

We consider a minimal, free action, φ, of the group ℤ^d on the Cantor set X, for d ≥ 1. We introduce the notion of small positive cocycles for such an action. We show that the existence of such cocycles allows the construction of finite Kakutani–Rohlin approximations to the action. In the case, d = 1, small positive cocycles always exist and the approximations provide the basis for the Bratteli–Vershik model for a minimal homeomorphism of X. Finally, we consider two classes of examples when d = 2 and show that such cocycles exist in both.

In this paper, we study the existence of an invariant foliation for a class of stochastic partial differential equations with a multiplicative white noise. This invariant foliation is used to trace the long term behavior of all solutions of these equations.

For cocycles with discrete time, we consider the notion of an exponential dichotomy in mean. This corresponds to replace the classical notion of an exponential dichotomy by the much weaker requirement that the same happens in mean with respect to some probability measure. We show that the exponential behavior in mean is robust, in the sense that it persists under sufficiently small linear perturbations.

We consider pinching cocycles taking values in the space of homeomorphisms of the circle over an hyperbolic base. Using the invariance principle of Malicet, we prove that the cocycles having nonzero exponents of contraction are dense. In this paper, we generalize some common notions an results known of linear cocycles and cocycles of diffeomorphisms, to the nonlinear non-differentiable case.

We introduce a theory of cyclic Kummer extensions of commutative rings for partial Galois extensions of finite groups, extending some of the well-known results of the theory of Kummer extensions of commutative rings developed by Borevich. In particular, we provide necessary and sufficient conditions to determine when a partial $n$-Kummerian extension is equivalent to either a radical or an $I$-radical extension, for some subgroup $I$ of the cyclic group ${\mathbb{Z}}_n$.