Parity sequences of the 3x+1 map on the 2-adic integers and Euclidean embedding
Dynamical Systems
2025-02-20 v6 Number Theory
Abstract
In this paper, we consider the one-to-one correspondence between a 2-adic integer and its parity sequence under iteration of the so-called "3x+1" map. First, we prove a new formula for the inverse transform. Next, we briefly review what is known about the induced automorphism and study its dynamics on the 2-adic integers. We find that it is ergodic on many small odd invariant sets, and that it has two odd cycles of period 2 in addition to its two odd fixed points. Finally, a plane embedding is presented, for which we establish affine self-similarity by using functional equations.
Keywords
Cite
@article{arxiv.1805.00133,
title = {Parity sequences of the 3x+1 map on the 2-adic integers and Euclidean embedding},
author = {Olivier Rozier},
journal= {arXiv preprint arXiv:1805.00133},
year = {2025}
}
Comments
30 pages, 2 figures. Various improvements. Added a reference to Monks et al (2013)