2010 seminar talk: Kakutani-von Neumann maps on simplexes

Talk held by Giovanni Panti (Udine) at the KGRC seminar on 2010-03-11.


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: [1]

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