摘要：AbstractWe extend the Collatz mapTto a new map T¯ whose trajectories converge to 1. This map gives rise to a fully binary tree which contains shifted copies of the Collatz tree. The new structure allows the simultaneous study of all important features of the conjecture, such as Collatz sequences, transition of parity vectors and the double indexed sequence of signs (−1)Tk(n). Furthermore, the binary tree leads to a direct system that can be used to study the convergence properties of the sequence of signs and it also extends the Collatz conjecture to new directions which are not directly related to the problem.
