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Escape rates and singular limiting distributions for intermittent maps with holes


Authors: Mark F. Demers and Bastien Fernandez
Journal: Trans. Amer. Math. Soc. 368 (2016), 4907-4932
MSC (2010): Primary 37C30, 37C40, 37D25, 37E05
DOI: https://doi.org/10.1090/tran/6481
Published electronically: December 3, 2015
MathSciNet review: 3456165
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Abstract: We study the escape dynamics in the presence of a hole of a standard family of intermittent maps of the unit interval with neutral fixed point at the origin (and finite absolutely continuous invariant measure). Provided that the hole (is a cylinder that) does not contain any neighborhood of the origin, the surviving volume is shown to decay at polynomial speed with time. The associated polynomial escape rate depends on the density of the initial distribution, more precisely, on its behavior in the vicinity of the origin. Moreover, the associated normalized push forward measures are proved to converge to the point mass supported at the origin, in sharp contrast to systems with exponential escape rate. Finally, a similar result is obtained for more general systems with subexponential escape rates, namely that the Cesàro limit of normalized push forward measures is typically singular, invariant and supported on the asymptotic survivor set.


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Additional Information

Mark F. Demers
Affiliation: Department of Mathematics, Fairfield University, Fairfield, Connecticut 06824

Bastien Fernandez
Affiliation: Centre de Physique Théorique, CNRS - Aix-Marseille Université - Université de Toulon, Campus de Luminy, 13288 Marseille CEDEX 9, France

DOI: https://doi.org/10.1090/tran/6481
Received by editor(s): November 6, 2013
Received by editor(s) in revised form: May 8, 2014, and May 30, 2014
Published electronically: December 3, 2015
Additional Notes: The first author was partially supported by NSF grant DMS 1101572
The second author was partially supported by EU FET Project No. TOPDRIM 318121
Article copyright: © Copyright 2015 American Mathematical Society