Activities This Week

Discrepancy is a measure of equidistribution for sequences of points. A bounded remainder set is a set with bounded discrepancy, that is, the number of times it is visited differs by the expected time only by a constant. We discuss dynamical, symbolic, and spectral approaches to the study of bounded remainder sets for Kronecker sequences. We consider in particular discrepancy in the setting of symbolic dynamics and we discuss the existence of bounded remainder sets for some families of zero entropy subshifts. Note that bounded discrepancy has also to do with the notion of bounded displacement to a lattice in the context of Delone sets. We focus on the case of Pisot parameters for toral translations and then show how to construct symbolic codings in terms of multidimensional continued fraction algorithms.
This is joint work with W. Steiner and J. Thuswaldner.

Consider a morphism of germs of Noetherian schemes, f: (X,x)-> (Y,y). When is it ‘stable’ under perturbations by higher order terms? I.e. when can such a perturbation be undone by a group action, e.g. by the local coordinate changes. This question has been extensively studied for real/complex analytic (or C^k) maps (k^n,o)-> (k^m,o). The idea is to reduce the orbit study, Gf, to the study of the tangent space, T_G f. The classical methods used vector field integration and infinite dimensional Lie groups, thus obstructing extensions to the zero/positive characteristic. During the last years we have developed a purely algebraic approach to this problem, extending the results to arbitrary characteristic. The key tool is the ‘Lie-type pair’. This is a group G, its would-be tangent space T_G, and certain maps between G, T_G, approximating the classical exponential/logarithm.

(joint work with G. Belitskii, A.F. Boix, G.M. Greuel.)