Quantifying the degree of average contraction of Collatz orbits

Timoteo Carletti, Duccio Fanelli

Research output: Working paper

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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
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 languageEnglish
PublisherNamur center for complex systems
Number of pages18
Publication statusSubmitted - 21 Dec 2016

Fingerprint

Contraction
Orbit
Modulo
Integer
Iterate
Congruence
Coarse-graining
Equilibrium Distribution
Transition Probability
Stochastic Systems
Markov chain
Fixed point
Distinct
Cycle
Invariant
Demonstrate
Class

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.
Carletti, Timoteo ; Fanelli, Duccio. / Quantifying the degree of average contraction of Collatz orbits. Namur center for complex systems, 2016.
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Carletti, T & Fanelli, D 2016 'Quantifying the degree of average contraction of Collatz orbits' Namur center for complex systems.

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

Namur center for complex systems, 2016.

Research output: Working paper

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AU - Fanelli, Duccio

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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.

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Carletti T, Fanelli D. Quantifying the degree of average contraction of Collatz orbits. Namur center for complex systems. 2016 Dec 21.