Bulletin of the Brazilian Mathematical Society

, Volume 33, Issue 2, pp 213–224

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

• A. V. Kontorovich
• Ya. G. Sinai
Article

## Abstract

The (3x + 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 (3x + 1)-Conjecture asks if for every x ε Π 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 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 (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 $$T(x) = \frac{{gx + h(gx)}} {{d^k }}$$, 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

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

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