A golden iterated map number system: Results and conjectures

Abstract

We construct a number system for representing numbers in [0, 1] that is based on iterations of an asymmetric tent map that incorporates the golden ratio. It is already known that in this system a number has a periodic representation if and only if it lies in [Formula: see text]. We investigate other aspects of this system such as non-uniquely representable numbers, an inherent semigroup structure, connections with Wythoff's game, related sequences of rational functions, and its connection with an iterative scheme reminiscent of paper folding analysis. Many details of the above-mentioned connections are conjectural.