Rare events for the Manneville–Pomeau map

Abstract

We prove a dichotomy for Manneville–Pomeau maps f:[0,1]→[0,1]: given any point ζ∈[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 Haydn (2014), and then to deal with the remaining points separately. The preimages of 0 are dealt with applying recent results in Aytaç (2015). 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.