[Paper Review] Extremes and extremal indices for level set observables on hyperbolic systems

Meagan Carney, Mark Holland+1 more|arXiv (Cornell University)|Sep 10, 2019

Mathematical Dynamics and Fractals39 references4 citations

TL;DR

This paper establishes extreme value laws and extremal indices for observables maximized on curves or submanifolds in hyperbolic dynamical systems, showing that clustering of exceedances arises from geometric self-intersection of submanifolds under dynamics, not periodicity. It derives nontrivial extremal indices for hyperbolic toral automorphisms, Sinai billiards, and coupled expanding maps, demonstrating that the extremal index depends on the alignment of the extremal set with stable/unstable manifolds.

ABSTRACT

Consider an ergodic measure preserving dynamical system $(T,X,μ)$, and an observable $ϕ:X o\mathbb{R}$. For the time series $X_n(x)=ϕ(T^{n}(x))$, we establish limit laws for the maximum process $M_n=\max_{k\leq n}X_k$ in the case where $ϕ$ is an observable maximized on a curve or submanifold, and $(T,X,μ)$ is a hyperbolic dynamical system. Such observables arise naturally in weather and climate applications. We consider the extreme value laws and extremal indices for these observables on Anosov diffeomorphisms, Sinai dispersing billiards and coupled expanding maps. In particular we obtain clustering and nontrivial extremal indices due to self intersection of submanifolds under iteration by the dynamics, not arising from any periodicity.

Motivation & Objective

• To establish extreme value laws for observables maximized on curves or submanifolds in hyperbolic dynamical systems.
• To investigate the origin of nontrivial extremal indices in the absence of periodicity, focusing on geometric self-intersection of submanifolds under dynamics.
• To extend extreme value theory beyond pointwise extremal sets to more complex geometric structures such as curves and planar sets.
• To analyze how the extremal index depends on the alignment of the extremal set with stable and unstable manifolds in hyperbolic systems.
• To address challenges in non-uniformly hyperbolic systems where the SRB measure is not equivalent to Lebesgue, particularly for fractal attractors and non-transverse extremal sets.

Proposed method

• The study employs extreme value theory for deterministic dynamical systems by analyzing the maximum process $ M_n = \max_{k \leq n} \phi(T^k(x)) $ for observables $ \phi $ maximized on a submanifold $ \mathcal{S} $.
• It uses the observable $ \phi(x) = -\log d_H(x, \mathcal{S}) $, where $ d_H $ is the Hausdorff distance to the extremal set $ \mathcal{S} $, to model level sets shrinking to $ \mathcal{S} $.
• The extremal index $ \theta $ is derived via the limit law $ \mu(M_n \leq u_n(\tau)) \to e^{-\theta \tau} $, with $ u_n(\tau) $ chosen such that $ n \mu(\phi > u_n) \to \tau $.
• For hyperbolic toral automorphisms and Sinai billiards, the extremal index is computed explicitly, showing $ \theta = 1 - \frac{1}{\lambda^2} $ when the extremal set aligns with unstable manifolds and contains a periodic orbit.
• The analysis incorporates transversality conditions between $ \mathcal{S} $ and global stable/unstable manifolds, and examines non-generic cases where alignment causes nontrivial $ \theta $.
• The paper extends results to coupled expanding maps and discusses the challenges in non-uniformly hyperbolic systems with fractal attractors and non-regular measures.

Experimental results

Research questions

• RQ1 What extreme value laws govern time series generated by observables maximized on curves in hyperbolic dynamical systems?
• RQ2 How does the extremal index behave when the extremal set is a smooth curve rather than a single point?
• RQ3 Can nontrivial extremal indices arise from geometric self-intersection of submanifolds under dynamics, independent of periodicity?
• RQ4 What is the dependence of the extremal index on the alignment of the extremal set with stable and unstable manifolds in hyperbolic systems?
• RQ5 How do extreme value laws change in non-uniformly hyperbolic systems where the SRB measure is not absolutely continuous with respect to Lebesgue measure?

Key findings

• For Arnold's cat map with $ \phi(x) = -\log d(x, L) $, where $ L $ is a line aligned with the unstable direction and contains a 2-periodic point, the extremal index is $ \theta = 1 - \frac{1}{\lambda^2} $, with $ \lambda $ the expansion rate.
• When the extremal set $ L $ is not aligned with stable or unstable directions, the extremal index is $ \theta = 1 $, indicating no clustering of exceedances.
• Numerical estimations for 10 realizations on Arnold's cat map show variation from theoretical $ \theta $ values within $ O(10^{-2}) $, confirming robustness.
• The extremal index depends on the geometric relationship between the extremal set and the global stable/unstable manifolds, not on periodicity alone.
• In systems like Sinai dispersing billiards and coupled expanding maps, nontrivial extremal indices emerge due to self-intersection of submanifolds under iteration, even without periodic orbits.
• For non-uniformly hyperbolic systems with fractal attractors, the existence of a GEV limit law is not guaranteed and depends on regular variation of $ \mu(\mathcal{S}_\epsilon) $ as $ \epsilon \to 0 $, requiring fine geometric and measure-theoretic analysis.