Abstract
A ‘next’ operator, σ, is built on the set R 1 =(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 R 1. In addition, the orbits {σn(α)} n ∈ Z are all dense in R 1 and are constituted by elements of the same arithmetical character: if α 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.
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Paradís, J., Viader, P. & Bibiloni, L. A Total Order in (0, 1] Defined Through a ‘Next’ Operator. Order 16, 207–220 (1999). https://doi.org/10.1023/A:1006441703404
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DOI: https://doi.org/10.1023/A:1006441703404