Escape rates and singular limiting distributions for intermittent maps with holes

Demers, Mark; Fernandez, Bastien

doi:10.1090/tran/6481

Escape rates and singular limiting distributions for intermittent maps with holes
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by Mark F. Demers and Bastien Fernandez PDF

Trans. Amer. Math. Soc. 368 (2016), 4907-4932 Request permission

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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