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

The (3x + 1)-Map, T, acts on the set, Π, of positive integers not divisible by 2 or 3. It is defined by , 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 , 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 and positive diffusion constant. This shows that it is natural to expect that typical trajectories return to the origin if and escape to infinity otherwise. Received: 18 April 2002