### Abstract

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.

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

Pages (from-to) | 445-468 |

Number of pages | 24 |

Journal | Bollettino dell'Unione Matematica Italiana |

Volume | 11 |

Issue number | 4 |

DOIs | |

Publication status | Published - 3 Oct 2017 |

### Fingerprint

### Keywords

- Collatz conjecture
- Number theory
- Markov process
- Ergodic dynamical systems

### Cite this

*Bollettino dell'Unione Matematica Italiana*,

*11*(4), 445-468. https://doi.org/10.1007/s40574-017-0145-x

}

*Bollettino dell'Unione Matematica Italiana*, vol. 11, no. 4, pp. 445-468. https://doi.org/10.1007/s40574-017-0145-x

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

Research output: Contribution to journal › Article

TY - JOUR

T1 - Quantifying the degree of average contraction of Collatz orbits

AU - Carletti, Timoteo

AU - Fanelli, Duccio

PY - 2017/10/3

Y1 - 2017/10/3

N2 - 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.

AB - 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.

KW - Collatz conjecture

KW - Number theory

KW - Markov process

KW - Ergodic dynamical systems

UR - http://www.scopus.com/inward/record.url?scp=85035759973&partnerID=8YFLogxK

U2 - https://doi.org/10.1007/s40574-017-0145-x

DO - https://doi.org/10.1007/s40574-017-0145-x

M3 - Article

VL - 11

SP - 445

EP - 468

JO - Bollettino dell'Unione Matematica Italiana

JF - Bollettino dell'Unione Matematica Italiana

SN - 1972-6724

IS - 4

ER -