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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 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.
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 |
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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 |
Keywords
- Collatz conjecture
- Number theory
- Markov process
- Ergodic dynamical systems
Fingerprint
Dive into the research topics of 'Quantifying the degree of average contraction of Collatz orbits'. Together they form a unique fingerprint.Projects
- 1 Finished
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PAI n°P7/19 - DYSCO: Dynamical systems, control and optimization (DYSCO)
WINKIN, J., Blondel, V., Vandewalle, J., Pintelon, R., Sepulchre, R., Vande Wouwer, A. & Sartenaer, A.
1/04/12 → 30/09/17
Project: Research
Activities
- 1 Participation in conference
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CCS 2016
Timoteo Carletti (Speaker)
22 Sep 2016Activity: Participating in or organising an event types › Participation in conference