Numerical estimate of infinite invariant densities: application to Pesin-type identity

TL;DR

This paper introduces a numerical method to estimate infinite invariant densities in weakly chaotic maps with unstable fixed points, enabling accurate calculation of subexponential trajectory separation rates.

Contribution

It proposes a novel approach to estimate infinite invariant densities using long-time limits of scaled normalizable densities, improving numerical accuracy and resolving recent misunderstandings.

Findings

The method accurately estimates infinite densities in weakly chaotic maps.

Exact calculation of lambda_alpha matches simulation results.

Clarifies misconceptions in the literature regarding invariant densities.

Abstract

Weakly chaotic maps with unstable fixed points are investigated in the regime where the invariant density is non-normalizable. We propose that the infinite invariant density of these maps can be estimated using as the long time limit of t^(1-alpha) rho(x, t), in agreement with earlier work of Thaler. Here rho(x, t) is the normalizable density of particles. This definition uniquely determines the infinite density and is a valuable tool for numerical estimations. We use this density to estimate the subexponential separation lambda_alpha of nearby trajectories. For a particular map introduced by Thaler we use an analytical expression for the infinite invariant density to calculate lambda_alpha exactly, which perfectly matches simulations without fitting. Misunderstanding which recently appeared in the literature is removed.