# Structure Theorem for (*d*, *g*, *h*)-Maps

## Abstract

The (3*x* + 1)-Map, *T*, acts on the set, Π, of positive integers not divisible by 2 or 3. It is defined by \(T(x) = \frac{{3x + 1}}
{{2^k }}\), where *k* is the largest integer for which *T* (*x*) is an integer. The (3*x* + 1)-Conjecture asks if for every *x* ε Π there exists an integer, *n*, such that *T* ^{ n } (*x*) = 1. The Statistical (3*x* + 1)-Conjecture asks the same question, except for a subset of Π 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 (3*x* + 1)-Conjecture may be true. In this paper, we present the analogous theorem for some generalizations of the (3*x* + 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 \(T(x) = \frac{{gx + h(gx)}}
{{d^k }}\), 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 \(\log g - \frac{d}
{{d - 1}}\) and positive diffusion constant. This shows that it is natural to expect that typical trajectories return to the origin if \(\log g - \frac{d}
{{d - 1}}\log d < 0\) and escape to infinity otherwise.

## Keywords

3*x*+ 1 Problem 3

*n*+ 1 Problem Collatz Conjecture Structure Theorem (

*d*,

*g*,

*h*)-Maps Brownian Motion

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