Abstract
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.
Original language | English |
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Pages (from-to) | 207-220 |
Journal | Order |
Volume | 16 |
Issue number | 3 |
DOIs | |
Publication status | Published - 1 Jan 1999 |
Keywords
- Pierce expansions
- Singular functions
- Total orders