Quantifying the degree of average contraction of Collatz orbits

TL;DR

This paper provides a quantitative analysis supporting the Collatz conjecture by examining fixed points, contraction behavior, and Markov chain models on mod8 classes, suggesting all orbits tend toward the known cycle.

Contribution

It introduces a novel Markov chain approach on mod8 classes to demonstrate orbit contraction, offering new evidence for the Collatz conjecture's validity.

Findings

Identified fixed points 1, 2, and 4 forming the attracting cycle.

Showed the Markov chain's stationary distribution induces orbit contraction.

Confirmed contraction behavior persists across different levels of coarse graining.

Abstract

We here elaborate on a quantitative argument to support the validity of the Collatz conjecture, also known as the (3x + 1) or Syracuse conjecture. The analysis is structured as follows. First, three distinct fixed points are found for the third iterate of the Collatz map, which hence organise in a period 3 orbit of the original map. These are 1, 2 and 4, the elements which define the unique attracting cycle, as hypothesised by Collatz. To carry out the calculation we write the positive integers in modulo 8 (mod8 ), obtain a closed analytical form for the associated map and determine the transitions that yield contracting or expanding iterates in the original, infinite-dimensional, space of positive integers. Then, we consider a Markov chain which runs on the reduced space of mod8 congruence classes of integers. The transition probabilities of the Markov chain are computed from the deterministic map, by employing a measure that is invariant for the map itself. Working in this setting, we demonstrate that the stationary distribution sampled by the stochastic system induces a contracting behaviour for the orbits of the deterministic map on the original space of the positive integers. Sampling the equilibrium distribution on the congruence classes mod8^m for any m, which amounts to arbitrarily reducing the degree of imposed coarse graining, returns an identical conclusion.