THE 3n+1-PROBLEM AND HOLOMORPHIC DYNAMICS

SIMON LETHERMAN, DIERK SCHLEICHER, AND REG WOOD

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 ↦ 4 ↦ 2 ↦ 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.