Efficient computation of statistical properties of intermittent dynamics

TL;DR

This paper introduces a new computational framework combining Abel functions, Euler-Maclaurin summation, and Chebyshev Galerkin methods to efficiently analyze ergodic properties of intermittent maps, even in challenging regimes.

Contribution

The paper presents a novel, effective numerical approach that achieves exponential convergence in computing statistical properties of intermittent maps, overcoming previous computational difficulties.

Findings

Achieves exponential convergence with polynomial computational effort.

Enables analysis of intermittent dynamics across all parameter regimes.

Provides accurate estimates of invariant measures and mean return times.

Abstract

Intermittent maps of the interval are simple and widely-studied models for chaos with slow mixing rates, but have been notoriously resistant to numerical study. In this paper we present an effective framework to compute many ergodic properties of these systems, in particular invariant measures and mean return times. The framework combines three ingredients that each harness the smooth structure of these systems' induced maps: Abel functions to compute the action of the induced maps, Euler-Maclaurin summation to compute the pointwise action of their transfer operators, and Chebyshev Galerkin discretisations to compute the spectral data of the transfer operators. The combination of these techniques allows one to obtain exponential convergence of estimates for polynomially growing computational outlay, independent of the order of the map's neutral fixed point. This enables numerical exploration of intermittent dynamics in all parameter regimes, including in the infinite ergodic regime.