Rare events for the Manneville-Pomeau map

Ana Cristina Moreira Freitas, Jorge Freitas, Mike Todd, Sandro Vaienti

Research output: Contribution to journalArticlepeer-review

11 Citations (Scopus)

Abstract

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 .
Original languageEnglish
Pages (from-to)3463-3479
Number of pages1717
JournalStochastic Processes and their Applications
Volume126
Issue number11
Early online date10 May 2016
DOIs
Publication statusPublished - Nov 2016

Keywords

  • Extreme Value Theory
  • Intermittent maps
  • Recurrence

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