Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Olivier Rozier

TL;DR

This work analyzes parity sequences under the $3x+1$ map by extending the dynamics to the 2-adic integers and encoding parity via a conjugacy $Q$ with the 2-adic shift. It provides two inverse-transform formulas for recovering integers from finite parity vectors and develops a rich 2-adic dynamical framework, including ergodicity results on invariant sets and explicit odd cycles. The paper then introduces a Euclidean embedding of the 2-adic dynamics into the plane, $\mathbf Q_{3x+1}=(X,Y)(\mathbb Z_2)$, and proves that this set is injective with a dense structure of rational points, culminating in an affine self-similarity picture that yields Hausdorff dimension $1$. Collectively, these results illuminate the deep connections between $3x+1$ parity sequences, 2-adic ergodic behavior, and fractal planar representations, with conjectures guiding future understanding of periodic orbits and rationality patterns.

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.

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

TL;DR

This work analyzes parity sequences under the map by extending the dynamics to the 2-adic integers and encoding parity via a conjugacy with the 2-adic shift. It provides two inverse-transform formulas for recovering integers from finite parity vectors and develops a rich 2-adic dynamical framework, including ergodicity results on invariant sets and explicit odd cycles. The paper then introduces a Euclidean embedding of the 2-adic dynamics into the plane, , and proves that this set is injective with a dense structure of rational points, culminating in an affine self-similarity picture that yields Hausdorff dimension . Collectively, these results illuminate the deep connections between parity sequences, 2-adic ergodic behavior, and fractal planar representations, with conjectures guiding future understanding of periodic orbits and rationality patterns.

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.

Paper Structure

This paper contains 12 sections, 17 theorems, 76 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Two formulae for the inverse transform
  3. Ultrametric extension
  4. The space of 2-adic integers
  5. Parity sequences
  6. Automorphism
  7. Functional equations
  8. Ergodicity
  9. Cycles
  10. The $3x+1$ set
  11. Euclidean embedding
  12. Self-similarity

Key Result

Lemma 1

(First formulation of the inverse transform) Let $S$ be a finite binary sequence $\left( s_0, s_1, \ldots, s_{j-1} \right)$ of length $j$. The set of integers $n$ for which $V_{j}(n) = S$ is given by the congruence class

Figures (2)

  • Figure 1: (a) Coverings of $\normalfont \textbf{Q}_{3x+1}$ made of $2^k$ squares of side length $2^{1-k}$ for $k=4$, 5, and 6, from left to right. (b) The set $\normalfont \textbf{Q}_{3x+1}$ with (green) line segments indicating the rational points from Table \ref{['tab:points']}, along with their respective parameter value.
  • Figure 2: (a-b) Identical parts of $\normalfont \textbf{Q}_{3x+1}$ through the affine transformations \ref{['eq:trans_even']} and \ref{['eq:trans_odd']} in (a) and (b) respectively. (c) Enlarged parts of $\normalfont \textbf{Q}_{3x+1}$ delimited by some of the boxes in (b), namely, $J_{k}^{2}$ for $k$ even and $I_{k} \times J_{k}$ for $k$ odd, with $2 \leq k \leq 7$.

Theorems & Definitions (53)

  • Definition 1
  • Definition 2
  • Lemma 1
  • proof
  • Example 1
  • Definition 3
  • Theorem 1
  • proof
  • Example 2
  • Corollary 1
  • ...and 43 more