### The 3n+1-problem and holomorphic dynamics

Simon Letherman, Dierk Schleicher, and Reg Wood
Source: Experiment. Math. Volume 8, Issue 3 (1999), 241-251.

#### 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.

First Page:
Primary Subjects: 37F10
Secondary Subjects: 11B83, 37F50
Full-text: Open access