The 3x+1 Periodicity Conjeture in $\mathbb{R}$

TL;DR

This paper investigates the 3x+1 conjecture within the 2-adic and 3-adic number systems, analyzing the properties of trajectories, conjugacies, and expansions, and establishing conditions for aperiodicity and cyclicity.

Contribution

It introduces new conditions relating digit frequency to aperiodicity in 2-adic integers and explores the behavior of the conjugacy map in real and p-adic contexts, providing novel insights into the conjecture.

Findings

Conjugacy maps aperiodic 2-adic integers to aperiodic ones under certain digit frequency conditions.

Identifies irrational numbers with specific aperiodic 2-adic expansions.

Discovers notable behaviors of orbits with Sturmian words as parity vectors.

Abstract

The $3x+1$ map $T$ is defined on the $2$-adic integers $\mathbb{Z}_2$ by $T(x)=x/2$ for even $x$ and $T(x)=(3x+1)/2$ for odd $x$. It is still unproved that under iteration of $T$ the trajectory of any rational $2$-adic integer is eventually cyclic. A $2$-adic integer is rational if and only if its representation with $1$'s and $0$'s is eventually periodic. We prove that the $3x+1$ conjugacy $\Phi$ maps aperiodic $v\in\mathbb{Z}_2$ onto aperiodic $2$-adic integers provided that $\underline{\lim}\;(\frac{h}{\ell})_{\ell=1}^{\infty} > \frac{\ln(2)}{\ln(3)}$ where $h$ is the number of $1$'s in the first $\ell$ digits of $v$ with the following constraint: if there is a rational $2$-adic integer with a non-cyclic trajectory, then necessarily $\underline{\lim}\;(\frac{h}{\ell})_{\ell=1}^{\infty}=\frac{\ln(2)}{\ln(3)}$. We study $\Phi$ as an infinite series in $\mathbb{R}$ and obtain negative irrational numbers for which we compute their aperiodic $2$-adic expansion. We find prominent behaviors of the orbit of $x$ taking Sturmian words as parity vector. We also found amazing results of the terms of $\Phi$ in $\mathbb{R}$. We define the $\ell$'th iterate of $T$ for $\ell\rightarrow \infty$ in the ring of $3$-adic integers and obtain positive irrational numbers for which we compute their aperiodic $3$-adic expansion.