Rare events for the Manneville-Pomeau map
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We prove a dichotomy for Manneville-Pomeau maps ƒ : [0, 1] → [0, 1] : given any point ζ ε [0, 1] , either the Rare Events Point Processes (REPP), counting the number of exceedances, which correspond to entrances in balls around ζ, converge in distribution to a Poisson process; or the point ζ is periodic and the REPP converge in distribution to a compound Poisson process. Our method is to use inducing techniques for all points except 0 and its preimages, extending a recent result [HWZ14], and then to deal with the remaining points separately. The preimages of 0 are dealt with applying recent results in [AFV14]. The point ζ = 0 is studied separately because the tangency with the identity map at this point creates too much dependence, which causes severe clustering of exceedances. The Extremal Index, which measures the intensity of clustering, is equal to 0 at ζ = 0 , which ultimately leads to a degenerate limit distribution for the partial maxima of stochastic processes arising from the dynamics and for the usual normalising sequences. We prove that using adapted normalising sequences we can still obtain non-degenerate limit distributions at ζ = 0 .
Freitas , A C M , Freitas , J , Todd , M & Vaienti , S 2016 , ' Rare events for the Manneville-Pomeau map ' Stochastic Processes and their Applications , vol 126 , no. 11 , pp. 3463-3479 . DOI: 10.1016/j.spa.2016.05.001
Stochastic Processes and their Applications
© 2016, Elsevier BV. This work is made available online in accordance with the publisher’s policies. This is the author created, accepted version manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at www.sciencedirect.com / https://dx.doi.org/10.1016/j.spa.2016.05.001
Funding: CMUP (UID/MAT/00144/2013), which is funded by FCT (Portugal) with national (MEC) and European structural funds through the programs FEDER, under the partnership agreement PT2020.
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