Quantifying the degree of average contraction of Collatz orbits

Timoteo Carletti, Duccio Fanelli

Résultats de recherche: Contribution à un journal/une revueArticle

Résumé

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.

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

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    title = "Quantifying the degree of average contraction of Collatz orbits",
    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.",
    keywords = "Collatz conjecture, Number theory, Markov process, Ergodic dynamical systems",
    author = "Timoteo Carletti and Duccio Fanelli",
    year = "2017",
    month = "10",
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    doi = "https://doi.org/10.1007/s40574-017-0145-x",
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    journal = "Bollettino dell'Unione Matematica Italiana",
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    Quantifying the degree of average contraction of Collatz orbits. / Carletti, Timoteo; Fanelli, Duccio.

    Dans: Bollettino dell'Unione Matematica Italiana, 03.10.2017.

    Résultats de recherche: Contribution à un journal/une revueArticle

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    AU - Fanelli,Duccio

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    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.

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    KW - Markov process

    KW - Ergodic dynamical systems

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