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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
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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

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References

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