Some Borel measures associated with the generalized Collatz mapping

K. Matthews

Colloquium Mathematicae (1992)

  • Volume: 63, Issue: 2, page 191-202
  • ISSN: 0010-1354

Abstract

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This paper is a continuation of a recent paper [2], in which the authors studied some Markov matrices arising from a mapping T:ℤ → ℤ, which generalizes the famous 3x+1 mapping of Collatz. We extended T to a mapping of the polyadic numbers ℤ ^ and construct finitely many ergodic Borel measures on ℤ ^ which heuristically explain the limiting frequencies in congruence classes, observed for integer trajectories.

How to cite

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Matthews, K.. "Some Borel measures associated with the generalized Collatz mapping." Colloquium Mathematicae 63.2 (1992): 191-202. <http://eudml.org/doc/210145>.

@article{Matthews1992,
abstract = {This paper is a continuation of a recent paper [2], in which the authors studied some Markov matrices arising from a mapping T:ℤ → ℤ, which generalizes the famous 3x+1 mapping of Collatz. We extended T to a mapping of the polyadic numbers $\widehat\{ℤ\}$ and construct finitely many ergodic Borel measures on $\widehat\{ℤ\}$ which heuristically explain the limiting frequencies in congruence classes, observed for integer trajectories.},
author = {Matthews, K.},
journal = {Colloquium Mathematicae},
keywords = {topology of congruence classes; Collatz mapping; polyadic numbers; ergodic sets; ergodic measures},
language = {eng},
number = {2},
pages = {191-202},
title = {Some Borel measures associated with the generalized Collatz mapping},
url = {http://eudml.org/doc/210145},
volume = {63},
year = {1992},
}

TY - JOUR
AU - Matthews, K.
TI - Some Borel measures associated with the generalized Collatz mapping
JO - Colloquium Mathematicae
PY - 1992
VL - 63
IS - 2
SP - 191
EP - 202
AB - This paper is a continuation of a recent paper [2], in which the authors studied some Markov matrices arising from a mapping T:ℤ → ℤ, which generalizes the famous 3x+1 mapping of Collatz. We extended T to a mapping of the polyadic numbers $\widehat{ℤ}$ and construct finitely many ergodic Borel measures on $\widehat{ℤ}$ which heuristically explain the limiting frequencies in congruence classes, observed for integer trajectories.
LA - eng
KW - topology of congruence classes; Collatz mapping; polyadic numbers; ergodic sets; ergodic measures
UR - http://eudml.org/doc/210145
ER -

References

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  1. [1] P. Billingsley, Ergodic Theory and Information, Wiley, New York 1965. Zbl0141.16702
  2. [2] R. N. Buttsworth and K. R. Matthews, On some Markov matrices arising from the generalized Collatz mapping, Acta Arith. 55 (1990), 43-57. Zbl0653.10004
  3. [3] K. L. Chung, Markov Chains, Springer, Berlin 1960. Zbl0122.36601
  4. [4] K. R. Matthews and A. M. Watts, A generalization of Hasse's generalization of the Syracuse algorithm, Acta Arith. 43 (1984), 167-175. Zbl0479.10006
  5. [5] K. R. Matthews and A. M. Watts, A Markov approach to the generalized Syracuse algorithm, ibid. 45 (1985), 29-42. Zbl0521.10008
  6. [6] K. R. Parthasarathy, Probability Measures on Metric Spaces, Academic Press, New York 1967. Zbl0153.19101
  7. [7] M. Pearl, Matrix Theory and Finite Mathematics, McGraw-Hill, New York 1973. 
  8. [8] A. G. Postnikov, Introduction to Analytic Number Theory, Amer. Math. Soc., Providence, R.I., 1988. Zbl0641.10001
  9. [9] A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar. 8 (1957), 477-493. Zbl0079.08901

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