Ratio and Partition are Revealed in Proposed Graph on Reduced Collatz Dynamics

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Collatz conjecture is also known as 3X+1 problem. The original dynamics is occurred 3x+1 and x/2 during the process from any given starting integer to 1. We but propose to study reduced dynamics, which are occurred (3*x+1)/2 and x/2 during the process from a starting integer to the first integer less than the starting integer, because reduced dynamics is the building blocks of original dynamics and x/2 always follows 3*x+1. We thus propose a graph (directed binary tree) to represent all possible reduced dynamics, in which (3*x+1)/2 is denoted as I with right directed edge and x/2 is denoted as O with down directed edge. The graph can show some inner laws in reduced dynamics directly and visually as follows: (1) The ratio, which is the count of x/2 over the count of (3*x+1)/2, is always larger than a constant value, namely, ln1.5/ln2. (2) The regular partition of integers whose reduced dynamics equals a path consisting of I and/or O can also be observed. That is, all integers are partitioned regularly in the graph. Given any positive integer x that is i module 2^t (i is an odd integer), the first t computations in terms of I or O can be determined. If current x, which is computed after t (t is greater or equal to 2) computations of I or O, is less than x, then reduced dynamics is obtained or available. Otherwise, the residue class of x (namely, i module 2^t) is partitioned further into two halves (namely, i module 2^t+1 and i+2^t module 2^t+1), and either half presents I or O in the forthcoming (t+1)-th computation. We finally propose an algorithm that takes as input reduced dynamics and outputs a residue class who presents this reduced dynamics.