The Structure of the Route to the Period-three Orbit in the Collatz Map

Abstract: This study analyzes the Collatz map through nonlinear dynamics. By embedding integers in Sharkovsky's ordering, we show that odd initial values suffice for full dynamical characterization. We introduce ``direction phases'' to partition iterations into upward and downward phases, and derive a recursive function family parameterized by upward phase counts. Consequently, a logarithmic scaling law between iteration steps and initial values is revealed, demonstrating finite-time convergence to the period-three orbit. Moreover, we establish the equivalence of the Collatz map to a binary shift map, whose ergodicity guarantees universal convergence to attractors, providing additional support for convergence. Furthermore, we identify that basins of attraction follow power-law distributions and find that odd numbers classified by upward phases follow Gamma statistics. These results offer valuable insights into the dynamics of discrete systems and their connections to number theory.