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 |
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Keywords
- Collatz conjecture
- Number theory
- Markov process
- Ergodic dynamical systems
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Quantifying the degree of average contraction of Collatz orbits. / Carletti, Timoteo; Fanelli, Duccio.
In: Bollettino dell'Unione Matematica Italiana, Vol. 11, No. 4, 03.10.2017, p. 445-468.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 -