Kakutani-von Neumann maps on simplexes

TL;DR

This paper generalizes Kakutani-von Neumann maps to higher-dimensional simplexes using n-dimensional tent maps, resulting in piecewise-linear, uniquely ergodic transformations that uniformly distribute orbits and enumerate rational points.

Contribution

It introduces an n-dimensional generalization of Kakutani-von Neumann maps using tent maps, expanding their applicability to arbitrary dimensions.

Findings

Maps are piecewise-linear bijections with uniformly distributed orbits.

The transformations are uniquely ergodic with respect to Lebesgue measure.

Provides an enumeration of rational points via the orbit of a vertex.

Abstract

A Kakutani-von Neumann map is the push-forward of the group rotation (Z_2,+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. In this paper 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.