Rigorous pointwise approximations for invariant densities of nonuniformly expanding maps

TL;DR

This paper develops a rigorous Ulam-type discretization method to approximate invariant densities of nonuniformly expanding interval maps with a neutral fixed point, providing explicit convergence rates.

Contribution

It introduces a novel pointwise approximation scheme with proven convergence rates for invariant densities in nonuniformly expanding maps.

Findings

Convergence rate of $C^*\frac{\ln m}{m}$ for the approximation

Explicit computable constant $C^*$ for the convergence

Effective method for invariant density approximation in complex maps

Abstract

We use an Ulam-type discretization scheme to provide pointwise approximations for invariant densities of interval maps with a neutral fixed point. We prove that the approximate invariant density converges pointwise to the true density at a rate $C^*\cdot\frac{\ln m}{m}$, where $C^*$ is a computable fixed constant and $m^{-1}$ is the mesh size of the discretization.