## Real $3x+1$

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- by Michał Misiurewicz and Ana Rodrigues
- Proc. Amer. Math. Soc.
**133**(2005), 1109-1118 - DOI: https://doi.org/10.1090/S0002-9939-04-07696-8
- Published electronically: October 15, 2004
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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 $T_1$.## References

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## Bibliographic Information

**Michał Misiurewicz**- Affiliation: Department of Mathematical Sciences, IUPUI, 402 N. Blackford Street, Indianapolis, Indiana 46202-3216
- MR Author ID: 125475
- 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
- 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: https://doi.org/10.1090/S0002-9939-04-07696-8
- MathSciNet review: 2117212