Generalized Cantor functions: random function iteration

Armstrong, Jordan; Schaubroeck, Lisbeth

doi:10.2140/involve.2020.13.281

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Generalized Cantor functions: random function iteration

Jordan Armstrong and Lisbeth Schaubroeck

Vol. 13 (2020), No. 2, 281–299

DOI: 10.2140/involve.2020.13.281

Abstract

We provide a generalization of the classical Cantor function. One characterization of the Cantor function is generated by a sequence of real numbers that starts with a seed value and at each step randomly applies one of two different linear functions. The Cantor function is defined as the probability that this sequence approaches infinity. We generalize the Cantor function to instead use a set of any number of linear functions with integer coefficients. We completely describe the resulting probability function and give a full explanation of which intervals of seed values lead to a constant probability function value.

Keywords

iteration, cantor function, sequence, devil's staircase

Mathematical Subject Classification 2010

Primary: 26A18

Milestones

Received: 13 May 2019

Revised: 11 December 2019

Accepted: 23 December 2019

Published: 30 March 2020

Communicated by Michael Dorff

Authors

Jordan Armstrong
Department of Mathematical Sciences U.S. Air Force Academy Air Force Academy, CO United States

Lisbeth Schaubroeck
Department of Mathematical Sciences U.S. Air Force Academy Air Force Academy, CO United States

Vol. 13, No. 2, 2020