Real 3𝑥+1

Misiurewicz, Michał; Rodrigues, Ana

doi:10.1090/S0002-9939-04-07696-8

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 $T_1$.

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