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

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 Tⁿ (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.