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A total order in [0,1] defined through a 'next' operator

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Abstract

A `next' operator, s, 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. Besides, the orbits {sn(a)}n 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 a's orbit are algebraic of degree k; if a 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.

Suggested Citation

  • Jaume Paradís & Pelegrí Viader & Lluís Bibiloni, 1998. "A total order in [0,1] defined through a 'next' operator," Economics Working Papers 266, Department of Economics and Business, Universitat Pompeu Fabra.
    • Handle: RePEc:upf:upfgen:266
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    Keywords

    Total orders; pierce series; singular functions;
    All these keywords.

    JEL classification:

    • C00 - Mathematical and Quantitative Methods - - General - - - General

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