Parity sequences of the 3x+1 map on the 2-adic integers and Euclidean embedding

TL;DR

This paper explores the dynamics of the 3x+1 map on 2-adic integers through parity sequences, providing new formulas, analyzing ergodic properties, and presenting a self-similar Euclidean embedding.

Contribution

It introduces a new inverse formula, studies the automorphism's ergodic behavior, and constructs a self-similar Euclidean embedding of the 3x+1 dynamics.

Findings

Proved a new formula for the inverse transform of parity sequences.

Identified ergodic behavior on small odd invariant sets.

Established affine self-similarity in a Euclidean plane embedding.

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.