Pseudoperiodicity and the $3x+1$ Conjugacy Function

TL;DR

This paper explores the 3x+1 function's extension to 2-adic integers, investigating pseudoperiodicity and fixed points of the conjugacy function, providing evidence for the conjecture about its fixed points.

Contribution

It introduces the concept of pseudoperiodicity in the 2-adic extension of the 3x+1 function and proves the existence of nontrivial pseudoperiodic sequences, supporting the fixed point conjecture.

Findings

Identified nontrivial pseudoperiodic sequences in 2-adic integers.

Proved three sequences of farPoints are finitely pseudoperiodic.

Provided evidence supporting the $\

Abstract

The 3x+1 function T is defined on the positive integers by $T(x) = \frac{3x+1}{2}$ for x odd and $T(x) = \frac{x}{2}$ for x even. The function T has a natural extension to the 2-adic integers, and there is a continuous function $\Phi$ which conjugates T to the 2-adic shift map $\sigma$. Bernstein and Lagarias conjectured that -1 and 1/3 are the only odd fixed points of $\Phi$. In this paper we investigate periodicity associated with $\Phi$, a property of the map which is a natural extention of solenoidality. We use it to show that there are nontrivial infinite families of 2-adics that are not fixed points of $\Phi$. In particular, we prove that three sequences of farPoints of 2-adic integers are finitely pseudoperiodic, providing more evidence supporting the $\Phi$ Fixed Point Conjecture.