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Extreme Value Theory with Spectral Techniques: application to a simple attractor

Jason Atnip, Nicolai Haydn, Sandro Vaienti

TL;DR

This work develops a spectral perturbation framework to study extreme value laws for dynamical systems, focusing on a bi-dimensional baker map as a concrete prototype. By constructing anisotropic Banach spaces and proving quasi-compactness of the transfer operator, the authors apply Keller–Liverani perturbation theory to derive a precise limiting law for maximal observables and to identify the extremal index via the geometry of shrinking targets. They show that, under appropriate scaling, the maximum converges to a Gumbel law with index $\\theta$, with nontrivial CP-type cluster behavior near periodic points depending on the metric and target shape. The analysis extends to counting visits in shrinking targets, yielding compound Poisson statistics and linking EVT to hitting-time theory, while also outlining generalizations to other observables and broader dynamical settings.

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.

Extreme Value Theory with Spectral Techniques: application to a simple attractor

TL;DR

This work develops a spectral perturbation framework to study extreme value laws for dynamical systems, focusing on a bi-dimensional baker map as a concrete prototype. By constructing anisotropic Banach spaces and proving quasi-compactness of the transfer operator, the authors apply Keller–Liverani perturbation theory to derive a precise limiting law for maximal observables and to identify the extremal index via the geometry of shrinking targets. They show that, under appropriate scaling, the maximum converges to a Gumbel law with index , with nontrivial CP-type cluster behavior near periodic points depending on the metric and target shape. The analysis extends to counting visits in shrinking targets, yielding compound Poisson statistics and linking EVT to hitting-time theory, while also outlining generalizations to other observables and broader dynamical settings.

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.

Paper Structure

This paper contains 25 sections, 8 theorems, 157 equations, 3 figures.

Table of Contents

  1. Introduction
  2. A simple example: the generalized baker's map
  3. The map
  4. The functional space
  5. The spectral approach for EVT
  6. Formulation of the problem
  7. Target sets and the space $\mathcal{B}$
  8. ${\bf 1}_{B_{n}^c}$ is in the Banach space $\mathcal{B}$
  9. ${\bf 1}_{B^{(k)}}h\in \mathcal{B}$
  10. The perturbative approach
  11. Proof of A2
  12. Weak norm
  13. Strong stable norm
  14. Strong unstable norm
  15. Proof of A4
  16. ...and 10 more sections

Key Result

Lemma 5.1

There exists two constants $\hat{C}_1, \hat{C}_2$ independent of $n$ such that

Figures (3)

  • Figure 1: Action of the baker's map on the unit square. The lower part of the square is mapped to the left part and the upper part is mapped to the right part.
  • Figure 2: Computation of the extremal index around periodic point with the Euclidean metric. The vertical line is an unstable manifold. We should compute the green area inside the circle.
  • Figure 3: Computation of the extremal index around periodic point with the $l^{\infty}$ metric. We should compute the green area inside the square.

Theorems & Definitions (18)

  • Remark 4.1
  • Remark 4.2
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • proof
  • Remark 5.3
  • Remark 5.4
  • Proposition 6.1
  • Lemma 7.1
  • ...and 8 more