Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-24T14:57:43.806Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

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]Downarowicz, T., Mauldin, R. D. & Monticino, M. G.. Exchangeable trees and random distributions. Preprint.Google Scholar
[2]Dubins, L. E. & Freedman, D. A.. Random distribution functions. In: Proceedings, Fifth Berkeley Symp. on Math. Statistics and Probability. LeCam, L. M. and Neyman, J., eds, pp. 183–214. University of California Press, Berkeley/Los Angeles, 1967.Google Scholar
[3]Graf, S., Mauldin, R. D. & Williams, S. C.. Random homeomorphisms. Adv. Math. 60 (1986), 239–359.CrossRefGoogle Scholar
[4]Graf, S., Novak, E. & Papageorgiou, A.. Bisection is not optimal on the average. Numer. Math. 55 (1989), 481–491.CrossRefGoogle Scholar
[5]Novak, E.. Deterministic and Stochastic Error Bounds in Numerical Analysis. Springer Lecture Notes in Mathematics 1349. Springer-Verlag, New York, 1989.Google Scholar
[6]Palis, J. Jr & de Melo, W.. Geometric Theory of Dynamical Systems. Springer-Verlag, New York, 1980.Google Scholar