Quantifying the degree of average contraction of Collatz orbits

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
defi?ne 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,
in?finite-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.