An Explicit Near-Conjugacy Between the Collatz Map and a Circle Rotation

TL;DR

This paper constructs an explicit logarithmic transformation that nearly conjugates the Collatz map to an irrational circle rotation, revealing a new geometric perspective and bounded error structure in Collatz dynamics.

Contribution

It introduces a precise near-conjugacy between the Collatz map and a circle rotation, with explicit bounds and a detailed analysis of the error term, advancing the understanding of Collatz dynamics.

Findings

Transformation maps Collatz to a circle rotation with bounded error

Numerical evidence confirms the bounds and boundedness of errors up to 10^{12}

Provides a geometric framework for analyzing Collatz trajectories

Abstract

We introduce an explicit logarithmic transformation $T(x) = \{\log_6(x + 1/5)\}$ under which the Collatz map becomes a rigid circle rotation by the irrational angle \(\alpha = \log_6 3\), perturbed by a uniformly bounded error term. We prove that for all positive integers \(x\), $T(C(x)) = T(x) + \alpha + \varepsilon(x) \pmod{1}$, where \(|\varepsilon(x)| \le 0.2749\) and \(\varepsilon(x) = O(1/x)\) as \(x \to \infty\). We derive the transformation from an exact functional equation linking the even and odd branches of the Collatz map, explain the arithmetic origin of the parameters \(6\) and \(1/5\), and analyse the structure of the resulting error term. Extensive numerical computations up to \(10^{12}\) confirm the sharpness of the bounds and show that cumulative errors remain uniformly bounded along all tested trajectories. While this near-conjugacy does not by itself resolve the Collatz conjecture, it provides a concrete and quantitative dynamical framework that clarifies the geometric structure underlying the Collatz iteration and may be useful in further analytical or experimental investigations of Collatz-type systems.