Some Borel measures associated with the generalized Collatz mapping

• Autorzy: K. R. Matthews
• Języki publikacji: EN

Abstrakty

EN

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.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 1992
• Tom: 63
• Numer: 2
• Strony: 191-202

Daty

wydano

1992

otrzymano

1990-12-07

poprawiono

1991-01-11

Twórcy

autor

K. R. Matthews

• Department of Mathematics, University of Queensland, St. Lucia, Brisbane 4072, Australia

Bibliografia

• [1] P. Billingsley, Ergodic Theory and Information, Wiley, New York 1965.
• [2] R. N. Buttsworth and K. R. Matthews, On some Markov matrices arising from the generalized Collatz mapping, Acta Arith. 55 (1990), 43-57.
• [3] K. L. Chung, Markov Chains, Springer, Berlin 1960.
• [4] K. R. Matthews and A. M. Watts, A generalization of Hasse's generalization of the Syracuse algorithm, Acta Arith. 43 (1984), 167-175.
• [5] K. R. Matthews and A. M. Watts, A Markov approach to the generalized Syracuse algorithm, ibid. 45 (1985), 29-42.
• [6] K. R. Parthasarathy, Probability Measures on Metric Spaces, Academic Press, New York 1967.
• [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.
• [9] A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar. 8 (1957), 477-493.