Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-28T03:38:38.200Z Has data issue: false hasContentIssue false
  • English
  • Français

Ergodic properties of a dynamical system arising from percolation theory

Published online by Cambridge University Press:  19 September 2008

Cor Kraaikamp  and
Ronald Meester
Cor Kraaikamp
Affiliation:
Delft University of Technology, Department of Mathematics, Mekelweg 4, 2628 CD Delft, The Netherlands
Ronald Meester
Affiliation:
University of Utrecht, Department of Mathematics, P.O. Box 80.010, 3508 TA Utrecht, The Netherlands
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the following dynamical system: take a d-dimensional real vectorwith positive coordinates. Now keep the smallest coordinate and subtract this one from the others, and iterate this process. When the starting vector is x we denote by xn the result after n iterations. It is shown that for almost all x, limn→∞xn ≠ 0 (the null vector). This is shown to be equivalent to the conjectured finiteness of an algorithm which produces the critical probability in a certain dependent percolation model.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 15 , Issue 4 , August 1995 , pp. 653 - 661
Copyright
Copyright © Cambridge University Press 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Billingsley, P.. Ergodic theory and information. Wiley Series in Probability and Mathematical Statistics. Wiley, Chichester, 1965.Google Scholar
[2]Meester, R.W.J.. An algorithm for calculating critical probabilities and percolation functions in percolation models defined by rotations. Ergod. Th. & Dynam. Sys. 9 (1989), 495–509.CrossRefGoogle Scholar
[3]Meester, R.W.J. and Nowicki, T.. Infinite clusters and critical values in two-dimensional circle percolation, Isr. J. Math. 68 (1989), 63–81.CrossRefGoogle Scholar
[4]R, A.ényi. Representations of real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungary 8 (1957), 477–493.Google Scholar
[5]Schweiger, F.. Invariant measures for maps of continued fraction type. J. Number Theory 39 (1991), 162–174.CrossRefGoogle Scholar