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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Real $3x+1$
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by Michał Misiurewicz and Ana Rodrigues PDF
Proc. Amer. Math. Soc. 133 (2005), 1109-1118 Request permission


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
  • Michał Misiurewicz
  • Affiliation: Department of Mathematical Sciences, IUPUI, 402 N. Blackford Street, Indianapolis, Indiana 46202-3216
  • MR Author ID: 125475
  • Email:
  • Ana Rodrigues
  • Affiliation: Universidade do Minho, Escola de Ciencias, Departamento de Matematica, Campus de Gualtar, 4710-057 Braga, Portugal
  • Email:
  • 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
  • © Copyright 2004 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 133 (2005), 1109-1118
  • MSC (2000): Primary 37B05; Secondary 20M20, 37C25, 11B83
  • DOI:
  • MathSciNet review: 2117212