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

Jaume Paradís (jaume.paradis@upf.edu), Pelegrí Viader (pelegri.viader@upf.edu) and Lluís Bibiloni
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Pelegrí Viader: https://www.upf.edu/web/econ/faculty/-/asset_publisher/6aWmmXf28uXT/persona/id/3418630

Economics Working Papers from Department of Economics and Business, Universitat Pompeu Fabra

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.

Keywords: Total orders; pierce series; singular functions (search for similar items in EconPapers)
JEL-codes: C00 (search for similar items in EconPapers)
Date: 1998-01
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