A dynamical approach towards Collatz conjecture

TL;DR

This paper explores the Collatz conjecture through a dynamical systems perspective, analyzing the iterative map in binary form to estimate growth, identify periodic orbits, and provide new insights into the problem.

Contribution

It introduces a novel dynamical approach using binary iterative maps to study the Collatz conjecture, including new estimations and characterization of periodic orbits.

Findings

New estimation of the growth of iterated numbers

Identification of characteristic periods for periodic orbits

Insight into the dynamical structure of the Collatz map

Abstract

The present work focuses on the study of the renowned Collatz conjecture, also known as the $3x +1$ problem. The distinguished analysis approach lies on the dynamics of an iterative map in binary form. A new estimation of the enlargement of iterated numbers is given. Within the associated iterative map, characteristic periods for periodic orbits are identified.