Spectral degeneracy and escape dynamics for intermittent maps with a hole

TL;DR

This paper investigates the metastability and escape dynamics of intermittent maps with small holes, analyzing eigenfunctions and eigenvalues of transfer operators, and explaining numerical behaviors observed with Ulam's method.

Contribution

It provides a rigorous analysis of how small perturbations near an intermittent fixed point affect eigenvalues and eigenfunctions, linking spectral properties to metastability and numerical methods.

Findings

Convergence of ACCIMs to the ACIM as neighborhood shrinks

Precise scaling laws for second eigenvalues under perturbation

L^1 convergence of eigenfunctions to the invariant measure

Abstract

We study intermittent maps from the point of view of metastability. Small neighbourhoods of an intermittent fixed point and their complements form pairs of almost-invariant sets. Treating the small neighbourhood as a hole, we first show that the absolutely continuous conditional invariant measures (ACCIMs) converge to the ACIM as the length of the small neighbourhood shrinks to zero. We then quantify how the escape dynamics from these almost-invariant sets are connected with the second eigenfunctions of Perron-Frobenius (transfer) operators when a small perturbation is applied near the intermittent fixed point. In particular, we describe precisely the scaling of the second eigenvalue with the perturbation size, provide upper and lower bounds, and demonstrate $L^1$ convergence of the positive part of the second eigenfunction to the ACIM as the perturbation goes to zero. This perturbation and associated eigenvalue scalings and convergence results are all compatible with Ulam's method and provide a formal explanation for the numerical behaviour of Ulam's method in this nonuniformly hyperbolic setting. The main results of the paper are illustrated with numerical computations.