Rare events for the Manneville-Pomeau map

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:126

Article

Moreira Freitas, Ana Cristina^1,2  Freitas, Jorge Milhazes^3,4  Todd, Mike⁵  Vaienti, Sandro^6,7 

[1] Univ Porto, Ctr Matemat, Rua Dr Roberto Frias, P-4200464 Oporto, Portugal

[2] Univ Porto, Fac Econ, Rua Dr Roberto Frias, P-4200464 Oporto, Portugal

[3] Univ Porto, Ctr Matemat, Rua Campo Alegre 687, P-4169007 Oporto, Portugal

[4] Univ Porto, Fac Ciencias, Rua Campo Alegre 687, P-4169007 Oporto, Portugal

[5] Univ St Andrews, Math Inst, St Andrews KY16 9SS, Fife, Scotland

[6] Aix Marseille Univ, CNRS, CPT, UMR 7332, F-13288 Marseille, France

[7] Univ Toulon & Var, CNRS, CPT, UMR 7332, F-83957 La Garde, France

关键词: Extreme Value Theory;    Intermittent maps;    Recurrence;   

DOI  :  10.1016/j.spa.2016.05.001

来源: Elsevier

【 摘 要 】

We prove a dichotomy for Manneville-Pomeau maps f : [0, 1] -> [0, 1]: given any point zeta is an element of[0, 1], either the Rare Events Point Processes (REPP), counting the number of exceedances, which correspond to entrances in balls around zeta converge in distribution to a Poisson process; or the point zeta 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 Haydn (2014), and then to deal with the remaining points separately. The preimages of 0 are dealt with applying recent results in Aytac (2015). The point zeta = 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 zeta = 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 zeta = 0. (C) 2016 Elsevier B.V. All rights reserved.

【 授权许可】

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