A Total Order in (0, 1] Defined Through a 'Next' Operator

A 'next' operator, σ, is built on the set ℝ1 = (0, 1] - {1 - 1/e} defining a partial order that, with the help of the axiom of choice, can be extended to a total order in ℝ1. In addition, the orbits {σn(α)}n∈ℤ are all dense in ℝ1 and are constituted by elements of the same arithmetical character: if a is an algebraic irrational of degree k, all the elements in α's orbit are algebraic of degree k; if α is transcendental, all are transcendental. Moreover, the asymptotic distribution function of the sequence formed by the elements in any of the half-orbits is a continuous, strictly increasing, singular function very similar to the well-known Minkowski's?(·) function.