## Experimental Mathematics

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

### 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 $1\mapsto 4\mapsto 2\mapsto 1$. 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

**Permanent link to this document**

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

**Mathematical Reviews number (MathSciNet)**

MR1724157

**Zentralblatt MATH identifier**

1012.37028

**Subjects**

Primary: 37F10: Polynomials; rational maps; entire and meromorphic functions [See also 32A10, 32A20, 32H02, 32H04]

Secondary: 11B83: Special sequences and polynomials 37F50: Small divisors, rotation domains and linearization; Fatou and Julia sets

#### Citation

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