Real $3x+1$

Authors:
Michał Misiurewicz and Ana Rodrigues

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

Published electronically:
October 15, 2004

MathSciNet review:
2117212

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Abstract | References | Similar Articles | Additional Information

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

**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

Article copyright:
© Copyright 2004
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.