### 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

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.

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.

Original language | English |
---|---|

Publisher | Namur center for complex systems |

Number of pages | 18 |

Publication status | Submitted - 21 Dec 2016 |

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### Keywords

- number theory
- Dynamical systems
- Stochastic Processes
- Collatz conjecture

### Cite this

Carletti, T., & Fanelli, D. (2016).

*Quantifying the degree of average contraction of Collatz orbits*. Namur center for complex systems.