Collatz Attractors Are Space-Filling

Idriss J. Aberkane
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Idriss J. Aberkane: UNESCO-UniTwin Complex Systems Digital Campus, ECCE e-Lab, Strasbourg Université, CEDEX, 67081 Strasbourg, France

Mathematics, 2022, vol. 10, issue 11, 1-9

Abstract: The algebraic topology of Collatz attractors (or “Collatz Feathers”) remains very poorly understood. In particular, when pushed to infinity, is their set of branches discrete or continuous? Here, we introduce a fundamental theorem proving that the latter is true. For any odd x , we first define A x n as the set of all odd numbers with S y r ( x ) in their Collatz orbit and up to n more digits than x in base 2. We then prove lim n → ∞ | A x n | 2 n + c ≥ 1 with c > − 4 for all x and, in particular, c = 0 for x = 1 , which is a result strictly stronger than that of Tao 2019.

Keywords: dynamical system; 3 x + 1 problem; Collatz conjecture; discrete chaos; Furstenberg conjecture; discrete algebraic topology; chaos theory; chaotic cryptology (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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