Quantifying the degree of average contraction of Collatz orbits

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 mod $8^m$ for any m, which amounts to arbitrarily reducing the degree of imposed coarse graining, returns an identical conclusion.

Appendix A: Methods

1.1 Appendix A.1: A numerical test

The components of $\mathbf {P}_{stat}$ quantify the time spent by the system in each of the eight classes, in its long-lasting approach to equilibrium. To challenge this interpretative picture we compare in Fig. 2 the prediction for $\mathbf {P}_{stat}$ as obtained under the stochastic picture to the quasi-stationary distribution $\mathbf {P}^{num}_{stat}$ recorded numerically from a direct implementation of the deterministic Collatz map. By quasi-stationary distribution we here mean the frequency of visits of the eight classes, prior absorption to the Collatz cycle.

Fig. 2

The theoretical quasi-stationary distribution (filled blue circles) is compared to the homologous quantity obtained from a direct numerical implementation of the deterministic Collatz map. More specifically, starting from a generic integer n, we store the mod8 representation of the numbers obtained every 3 consecutive iterations of T, before the trajectory lands on the attractive, absorbing cycle. The frequency of visits for each of the allowed classes is then reported in the figure. Red crosses are obtained running the Collatz map for all integers in between 1 and $n_{max}=10^5$. Green squares refer to $n_{max}=10^7$. The blue horizontal lines are drawn as guideline for the reader, the solid line corresponds to 1 / 6 and the dashed one to 1 / 12. When increasing $n_{max}$, all entries of the quasi-stationary distribution shifts consistently, although imperceptibly, towards the theoretically predicted values. Observe that, for $n_{max} =10^7$, the largest value attained by the system is $M \simeq 6\times 10^{13}$ (colour figure online)