Experimental Mathematics

The 3n+1-problem and holomorphic dynamics

Simon Letherman, Dierk Schleicher, and Reg Wood


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

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

First available in Project Euclid: 9 March 2003

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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

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