### Abstract

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

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

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**Quantifying the degree of average contraction of Collatz orbits.** / Carletti, Timoteo; Fanelli, Duccio.

Research output: Working paper

TY - UNPB

T1 - Quantifying the degree of average contraction of Collatz orbits

AU - Carletti, Timoteo

AU - Fanelli, Duccio

PY - 2016/12/21

Y1 - 2016/12/21

N2 - We here elaborate on a quantitative argument to support the validity of theCollatz conjecture, also known as the (3x + 1) or Syracuse conjecture. Theanalysis is structured as follows. First, three distinct ?fixed points arefound for the third iterate of the Collatz map, which hence organise in aperiod 3 orbit of the original map. These are 1, 2 and 4, the elements whichdefi?ne the unique attracting cycle, as hypothesised by Collatz. To carry outthe calculation we write the positive integers in modulo 8 (mod8 ), obtain aclosed analytical form for the associated map and determine the transitionsthat yield contracting or expanding iterates in the original,in?finite-dimensional, space of positive integers. Then, we consider a Markovchain which runs on the reduced space of mod8 congruence classes of integers.The transition probabilities of the Markov chain are computed from thedeterministic map, by employing a measure that is invariant for the map itself.Working in this setting, we demonstrate that the stationary distributionsampled by the stochastic system induces a contracting behaviour for the orbitsof the deterministic map on the original space of the positive integers.Sampling the equilibrium distribution on the congruence classes mod8^m for anym, which amounts to arbitrarily reducing the degree of imposed coarse graining,returns an identical conclusion.

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

KW - number theory

KW - Dynamical systems

KW - Stochastic Processes

KW - Collatz conjecture

M3 - Working paper

BT - Quantifying the degree of average contraction of Collatz orbits

PB - Namur center for complex systems

ER -