Experimental Mathematics

The 3n+1-problem and holomorphic dynamics

Abstract

The 3n+1-problem is the following iterative procedure on the positive integers: the integer n maps to n/2 or 3n+1, depending on whether n is even or odd. It is conjectured that every positive integer will be eventually periodic, and the cycle it falls onto is $1\mapsto 4\mapsto 2\mapsto 1$. We construct entire holomorphic functions that realize the same dynamics on the integers and for which all the integers are in the Fatou set. We show that no integer is in a Baker domain (domain at infinity). We conclude that any integer that is not eventually periodic must be in a wandering domain.

Article information

Source
Experiment. Math., Volume 8, Issue 3 (1999), 241-251.

Dates
First available in Project Euclid: 9 March 2003

https://projecteuclid.org/euclid.em/1047262405

Mathematical Reviews number (MathSciNet)
MR1724157

Zentralblatt MATH identifier
1012.37028

Citation

Letherman, Simon; Schleicher, Dierk; Wood, Reg. The 3n+1-problem and holomorphic dynamics. Experiment. Math. 8 (1999), no. 3, 241--251. https://projecteuclid.org/euclid.em/1047262405