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Random circle homeomorphisms

Published online by Cambridge University Press:  19 September 2008

Tomasz Downarowicz ,
R. Daniel Mauldin  and
Tony T. Warnock
Tomasz Downarowicz
Affiliation:
Institute of Mathematics, Technical University, 50370 Wroclaw, Poland
R. Daniel Mauldin
Affiliation:
Mathematics Department, University of North Texas, Denton, Texas 76203, USA
Tony T. Warnock
Affiliation:
Los Alamos National Laboratory, Los Alamos, New Mexico, 87545, USA
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Abstract

We investigate the behaviour of random homeomorphisms of the circle induced by composing a random homeomorphism of the interval with a randomly chosen rotation. These maps and their iterates are a.s. singular and for each rational number r in [0,1) it is shown that there is a positive probability of obtaining a map with rotation number r. For a ‘canonical’ method of producing these maps, bounds on the probability of obtaining a fixed point are obtained. We estimate this probability via computer simulations in three different ways. Simulations are also carried out for two periods. It remains unknown for this method whether a rational rotation number is obtained a.s.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 12 , Issue 3 , September 1992 , pp. 441 - 458
Copyright
Copyright © Cambridge University Press 1992

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References

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