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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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Copyright information

© Sociedade Brasileira de Matemática 2002

Authors and Affiliations

  1. 1.Mathematics Department of Princeton UniversityPrincetonUSA

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