Letherman, Simon; Schleicher, Dierk; Wood, Reg The \(3n+1\)-problem and holomorphic dynamics. (English) Zbl 1012.37028 Exp. Math. 8, No. 3, 241-251 (1999). Summary: 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. Cited in 4 Documents MSC: 37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets 11B83 Special sequences and polynomials 37F50 Small divisors, rotation domains and linearization in holomorphic dynamics Keywords:\(3n+1\)-problem; entire holomorphic functions; Fatou set; wandering domain × Cite Format Result Cite Review PDF Full Text: DOI EuDML References: [1] Baker I. N., Ann. Acad. Sci. Fenn. Ser. A I Math. 1 (2) pp 277– (1975) [2] Bergweiler W., Bull. Amer. Math. Soc. (N.S.) 29 (2) pp 151– (1993) · Zbl 0791.30018 · doi:10.1090/S0273-0979-1993-00432-4 [3] Chamberland M., Dynam. Contin. Discrete Impuls. Systems 2 (4) pp 495– (1996) [4] Eremenko A. È., Algebra i Analiz 1 (3) pp 1– (1989) [5] Eremenko A. È., Ann. Inst. Fourier (Grenoble) 42 (4) pp 989– (1992) · Zbl 0735.58031 · doi:10.5802/aif.1318 [6] Halbeisen L., Acta Arith. 78 (3) pp 227– (1997) [7] Hofstadter D. R., Gödel, Escher, Bach: an eternal golden braid (1979) [8] Lagarias J. C., Amer. Math. Monthly 92 (1) pp 3– (1985) · Zbl 0566.10007 · doi:10.2307/2322189 [9] McMullen C. T., Adv. Math. 135 (2) pp 351– (1998) · Zbl 0926.30028 · doi:10.1006/aima.1998.1726 [10] de Melo W., One-dimensional dynamics, vol. 25, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) (1993) [11] Milnor J., Dynamics in one complex variable: introductory lectures (1999) · Zbl 0946.30013 [12] Roosendaal E., ”3x + 1 calculation algorithms and deep search results” (1999) [13] Wirsching G. J., The dynamical system generated by the 3n + 1 function (1998) · Zbl 0892.11002 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.