Algebra Seminar talk
2010-03-05
Giovanni Panti
Kakutani-von Neumann maps on simplexes
Abstract:
A Kakutani-von Neumann map is the push-forward of the group rotation (Z2,+1)
to a unit simplex via an appropriate topological quotient. The usual quotient
towards the unit interval is given by the base 2 expansion of real numbers,
which in turn is induced by the doubling map.
We replace the doubling map with an n-dimensional generalization of the tent map; this allows us to define Kakutani-von Neumann transformations in simplexes of arbitrary dimensions. The resulting maps are piecewise-linear bijections (not just mod 0 bijections), whose orbits are all uniformly distributed; in particular, they are uniquely ergodic w.r.t. the Lebesgue measure.
The forward orbit of a certain vertex provides an enumeration of all points in the simplex having dyadic coordinates, and this enumeration can be translated via the n-dimensional Minkowski function to an enumeration of all rational points. In the course of establishing the above results, we introduce a family of {+1,-1}-valued functions, constituting an n-dimensional analogue of the classical Walsh functions.
Paper reference:
http://arxiv.org/abs/1001.3324