（d，g，h）-映射的结构定理
Structure Theorem for (d,g,h)-Maps
摘要：
（3x+1）-映射T作用于不能被2或3整除的正整数的集合Pi。它定义为T（x）=（3x+1）/2^k，其中k是T（x）为整数的最大整数。（3x+1）-猜想询问是否对于Pi中的每个x都存在整数n，使得T^n（x）= 1。统计（3x+1）猜想提出了同样的问题，除了密度为1的Pi的子集。在\cite{sinai}中证明的结构定理表明，无穷大在某种意义上是一个排斥点，给出了一些理由来期望（3x+1）-猜想可能是正确的。本文给出了（3x+1）-映射的某些推广的类似定理，并推广了\cite{sinai}中的结果.我们考虑的推广由互质正整数d和g（g > d >= 2）和一个周期函数h（x）决定。映射T由公式T（x）=（gx+h（gx））/d^k定义，其中k也是T（x）为整数的最大整数。证明了（d，g，h）-映射的一个类似的结构定理，证明了对应于密度的概率分布收敛于漂移log（g）- d/（d-1）log（d）且扩散常数为正的Wiener测度.这表明，如果log（g）- d/（d-1）log（d）<0，则期望典型的仿射器返回到原点，否则逃逸到无穷大是很自然的。
英文摘要：
The (3x+1)-Map, T, acts on the set, Pi, 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 in Pi 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 Pi of density 1. The Structure Theorem proven in \cite{sinai} 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 \cite{sinai}. 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 trajectores return to the origin if log(g) - d/(d-1) log(d) <0 and escape to infinity otherwise.
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