Rare Events for the Manneville-Pomeau map
Ana Cristina Moreira Freitas, Jorge Milhazes Freitas, Mike Todd, and Sandro Vaienti

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 : given any point , either the Rare Events Point Processes (REPP), counting the number of exceedances, which correspond to entrances in balls around , converge in distribution to a Poisson process; or the point 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 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…
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