# Some Borel measures associated with the generalized Collatz mapping

Colloquium Mathematicae (1992)

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

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topMatthews, 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

top- [1] P. Billingsley, Ergodic Theory and Information, Wiley, New York 1965. Zbl0141.16702
- [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] K. L. Chung, Markov Chains, Springer, Berlin 1960. Zbl0122.36601
- [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] K. R. Matthews and A. M. Watts, A Markov approach to the generalized Syracuse algorithm, ibid. 45 (1985), 29-42. Zbl0521.10008
- [6] K. R. Parthasarathy, Probability Measures on Metric Spaces, Academic Press, New York 1967. Zbl0153.19101
- [7] M. Pearl, Matrix Theory and Finite Mathematics, McGraw-Hill, New York 1973.
- [8] A. G. Postnikov, Introduction to Analytic Number Theory, Amer. Math. Soc., Providence, R.I., 1988. Zbl0641.10001
- [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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