Rare Events for the Manneville-Pomeau map

TL;DR

This paper establishes a dichotomy for Rare Events Point Processes in Manneville-Pomeau maps, showing convergence to Poisson or compound Poisson processes depending on the point, with special treatment for the neutral fixed point at zero.

Contribution

The paper extends inducing techniques to analyze REPP convergence for all points in Manneville-Pomeau maps, including the challenging neutral fixed point at zero.

Findings

REPP converge to Poisson for non-periodic points

REPP converge to compound Poisson for periodic points

At zero, the extremal index is zero, leading to degenerate limits without adapted normalization

Abstract

We prove a dichotomy for Manneville-Pomeau maps $f:[0,1]\to [0, 1]$: given any point $\zeta\in [0,1]$, 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 by Haydn, Winterberg and Zweim\"uller, and then to deal with the remaining points separately. The preimages of 0 are dealt with applying recent results by Ayta\c{c}, Freitas and Vaienti. 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 $\zeta=0$.