[Paper Review] Extreme Value Theory with Spectral Techniques: application to a simple attractor

Jason Atnip, Nicolai Haydn+1 more|arXiv (Cornell University)|Feb 25, 2020

Mathematical Dynamics and Fractals16 references4 citations

TL;DR

This paper applies spectral techniques from transfer operator theory to extreme value statistics in dynamical systems, focusing on the baker's map. It derives a precise formula for the extremal index and establishes that the number of visits to small sets follows a compound Poisson distribution, demonstrating the method's effectiveness in non-stationary, higher-dimensional systems with non-trivial extremal sets.

ABSTRACT

We give a brief account of application of extreme value theory in dynamical systems by using perturbation techniques associated to the transfer operator. We will apply it to the baker's map and we will get a precise formula for the extremal index. We will also show that the statistics of the number of visits in small sets is compound Poisson distributed.

Motivation & Objective

• To extend extreme value theory in dynamical systems using spectral methods based on the transfer operator.
• To compute the extremal index for the baker’s map, a two-dimensional uniformly hyperbolic system with non-trivial geometry.
• To establish the limiting distribution of visit counts to small sets as compound Poisson, linking it to cluster structures in return times.
• To demonstrate the method’s applicability to systems with exponential decay of correlations and quasi-compact transfer operators, even in higher dimensions.
• To identify limitations of the spectral method in capturing full visit statistics, motivating alternative approaches.

Proposed method

• Utilizes perturbation theory of the transfer operator in anisotropic Banach spaces tailored to the baker’s map’s geometry.
• Employs the spectral gap of the transfer operator to derive the extremal index via the leading eigenvalue and spectral projection.
• Applies the connection between recurrence and extreme value laws established by Keller and Liverani, using the transfer operator’s quasi-compactness.
• Uses cylinder sets and dynamically relevant neighborhoods to estimate measures of small sets and their intersections under iterated maps.
• Relies on exact dimensionality of the SRB measure and volume estimates in Euclidean balls to derive scaling laws for visit probabilities.
• Compares the limiting return time distribution with the Pólya-Aeppli form, showing deviation and thus confirming a compound Poisson limit.

Experimental results

Research questions

• RQ1 Can spectral techniques of the transfer operator be used to compute the extremal index in a two-dimensional invertible system like the baker’s map?
• RQ2 Does the number of visits to small sets in the baker’s map follow a compound Poisson distribution, and if so, how is it characterized?
• RQ3 How does the extremal index derived via spectral methods relate to the expected cluster size in return time statistics?
• RQ4 What are the limitations of the spectral method in capturing full visit statistics, particularly in higher-dimensional systems?
• RQ5 Why does the limiting distribution of return times in the baker’s map fail to be Pólya-Aeppli, and what does this imply for the underlying stochastic process?

Key findings

• The extremal index for the baker’s map is computed as θ = 4/π × 2^{-kp} for periodic points of period p, with corrections of order O(2^{-2kp}).
• The limiting distribution of return times is not Pólya-Aeppli, indicating a compound Poisson distribution with non-geometric cluster size probabilities.
• The visit count statistics to small sets are shown to converge to a compound Poisson distribution, confirming the theoretical framework of [14].
• The spectral method successfully yields the extremal index and confirms consistency with the cluster size definition of the extremal index as the reciprocal of expected cluster length.
• The method fails to fully capture the visit statistics in the case of Euclidean balls around periodic points, where the limiting distribution deviates from Pólya-Aeppli.
• The analysis reveals that the choice of neighborhood (e.g., balls vs. cylinder sets) critically affects the limiting distribution, highlighting the need for dynamically relevant sets in spectral approaches.