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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Real $3x+1$


Authors: Michal Misiurewicz and Ana Rodrigues
Journal: Proc. Amer. Math. Soc. 133 (2005), 1109-1118
MSC (2000): Primary 37B05; Secondary 20M20, 37C25, 11B83
Published electronically: October 15, 2004
MathSciNet review: 2117212
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Abstract: The famous $3x+1$ problem involves applying two maps: $T_0(x)=x/2$ and $T_1(x)=(3x+1)/2$ to positive integers. If $x$ is even, one applies $T_0$, if it is odd, one applies $T_1$. The conjecture states that each trajectory of the system arrives to the periodic orbit $\{1,2\}$. In this paper, instead of choosing each time which map to apply, we allow ourselves more freedom and apply both $T_0$ and $T_1$independently of $x$. That is, we consider the action of the free semigroup with generators $T_0$ and $T_1$ on the space of positive real numbers. We prove that this action is minimal (each trajectory is dense) and that the periodic points are dense. Moreover, we give a full characterization of the group of transformations of the real line generated by $T_0$ and $T_1$.


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Additional Information

Michal Misiurewicz
Affiliation: Department of Mathematical Sciences, IUPUI, 402 N. Blackford Street, Indianapolis, Indiana 46202-3216
Email: mmisiure@math.iupui.edu

Ana Rodrigues
Affiliation: Universidade do Minho, Escola de Ciencias, Departamento de Matematica, Campus de Gualtar, 4710-057 Braga, Portugal
Email: anarodrigues@math.uminho.pt

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07696-8
PII: S 0002-9939(04)07696-8
Received by editor(s): November 26, 2003
Published electronically: October 15, 2004
Additional Notes: The authors were partially supported by NSF grant DMS 0139916. The second author thanks the hospitality of the Department of Mathematical Sciences of IUPUI
Communicated by: Michael Handel
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.