Quantifying the degree of average contraction of Collatz orbits

Timoteo Carletti, Duccio Fanelli

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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.
langue originaleAnglais
ÉditeurNamur center for complex systems
Nombre de pages18
Etat de la publicationSoumis - 21 déc. 2016

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  • naxys seminar

    Timoteo Carletti (Conférencier)

    25 oct. 2016

    Activité: Types de Participation ou d'organisation d'un événementParticipation à un atelier/workshop, un séminaire, un cours

  • CCS 2016

    Timoteo Carletti (Participant)

    19 sept. 201622 sept. 2016

    Activité: Types de Participation ou d'organisation d'un événementParticipation à une conférence, un congrès

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