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Transactions of the American Mathematical Society

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The loss of tightness of time distributions for homeomorphisms of the circle


Author: Zaqueu Coelho
Translated by:
Journal: Trans. Amer. Math. Soc. 356 (2004), 4427-4445
MSC (2000): Primary 37E05, 11A55, 37A50; Secondary 28D05, 60G55
DOI: https://doi.org/10.1090/S0002-9947-04-03386-0
Published electronically: February 4, 2004
MathSciNet review: 2067127
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Abstract: For a minimal circle homeomorphism $f$ we study convergence in law of rescaled hitting time point process of an interval of length $\varepsilon>0$. Although the point process in the natural time scale never converges in law, we study all possible limits under a subsequence. The new feature is the fact that, for rotation numbers of unbounded type, there is a sequence $\varepsilon_{n}$ going to zero exhibiting coexistence of two non-trivial asymptotic limit point processes depending on the choice of time scales used when rescaling the point process. The phenomenon of loss of tightness of the first hitting time distribution is an indication of this coexistence behaviour. Moreover, tightness occurs if and only if the rotation number is of bounded type. Therefore tightness of time distributions is an intrinsic property of badly approximable irrational rotation numbers.


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Additional Information

Zaqueu Coelho
Affiliation: Department of Mathematics, University of York, Heslington, York YO10 5DD, United Kingdom
Email: zc3@york.ac.uk

DOI: https://doi.org/10.1090/S0002-9947-04-03386-0
Keywords: Rotation numbers, limit laws, point processes, maps of the circle
Received by editor(s): August 31, 2001
Received by editor(s) in revised form: May 8, 2003
Published electronically: February 4, 2004
Article copyright: © Copyright 2004 American Mathematical Society

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