### Abstract

The (3x + 1)-Map, T, acts on the set, ∏, of positive integers not divisible by 2 or 3. It is defined by T (x) = 3x + 1/2^{k}, where k is the largest integer for which T (x) is an integer. The (3x + 1)-Conjecture asks if for every x ∈ ∏ there exists an integer, n, such that T^{n} (x) = 1. The Statistical (3x + 1)-Conjecture asks the same question, except for a subset of ∏ of density 1. The Structure Theorem proven in [S] shows that infinity is in a sense a repelling point, giving some reasons to expect that the (3x + 1)-Conjecture may be true. In this paper, we present the analogous theorem for some generalizations of the (3x + 1)-Map, and expand on the consequences derived in [S]. The generalizations we consider are determined by positive coprime integers, d and g, with g > d ≥ 2, and a periodic function, h (x). The map T is defined by the formula T (x) = gx + h (gx)/d^{k}, where k is again the largest integer for which T (x) is an integer. We prove an analogous Structure Theorem for (d, g, h)-Maps, and that the probability distribution corresponding to the density converges to the Wiener measure with the drift log g - d/d - 1 log d and positive diffusion constant. This shows that it is natural to expect that typical trajectories return to the origin if log g - d/d - 1 log d < 0 and escape to infinity otherwise.

Original language | English (US) |
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Pages (from-to) | 213-224 |

Number of pages | 12 |

Journal | Bulletin of the Brazilian Mathematical Society |

Volume | 33 |

Issue number | 2 |

DOIs | |

State | Published - Jul 2002 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Keywords

- (d, g, h)-Maps
- 3n + 1 Problem
- 3x + 1 Problem
- Brownian Motion
- Collatz Conjecture
- Structure Theorem

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## Cite this

*Bulletin of the Brazilian Mathematical Society*,

*33*(2), 213-224. https://doi.org/10.1007/s005740200010