Vol. 13, No. 2, 2020

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Generalized Cantor functions: random function iteration

Jordan Armstrong and Lisbeth Schaubroeck

Vol. 13 (2020), No. 2, 281–299
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