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

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.

💡 Research Summary

The paper presents a novel dynamical‑systems perspective on the classic Collatz (3 x + 1) problem. After a brief motivation that situates the Collatz map among a wide class of discrete dynamical systems exhibiting periodic, chaotic, and bifurcation phenomena, the authors introduce two key conceptual tools: “direction phases” and an equivalence to a binary shift (Bernoulli) map.

Direction phases. For any iterate Xₙ the authors define a binary indicator P↑(n)=+1 if Xₙ₊₁ > Xₙ (an upward step) and P↓(n)=‑1 if Xₙ₊₁ < Xₙ (a downward step). The total numbers of upward and downward steps before the orbit reaches the 4‑2‑1 cycle are denoted N↑ and N↓, respectively, with the total iteration count N = N↑ + N↓. By embedding the positive integers into Sharkovskiĭ’s ordering, the set of all integers is partitioned into the powers‑of‑two set B = {2ᵐ} and the mixed set D = {(2ⁿ + 1)·2ᵐ}. The authors argue that any even initial value eventually falls into B and thus can be ignored; the essential dynamics are captured by the odd set O = D ∩ ℕ⁺.

Recursive family Fₛ. For a given odd X₀ the number of upward phases is called s. The authors construct a family of recursive functions Fₛ(p, k₁,…,k_{s‑1}) that generate exactly those odd numbers that experience s upward steps. The base case s = 1 yields
 F₁(p) = (2^{2p} − 1)/3, p ≥ 2,
with iteration count N₁ = 1 + log₂(3X₀ + 1). For s = 2 they obtain
 F₂(p, k₁) = 2^{k₁}·F₁(p) − 1/3, k₁ ≥ 1,
and the total steps become N₂ = 2 + 2·log₂(3X₀ + 1) − log₂X₀. In the general case the recursion reads

 X₀ = Fₛ(p, k_{s‑1}) = 2^{k_{s‑1}}·F_{s‑1}(p, k_{s‑2}) − 1/3,

with

 k_{s‑1} = log₂(3X₀ + 1) − log₂X₀ ∈ ℕ.

Unfolding the recursion yields a closed‑form expression for the total number of iterations:

 Nₛ = s