On the regularity and approximation of invariant densities for random continued fractions

TL;DR

This paper investigates how small changes in random dynamical systems with spectral gaps affect their invariant densities, providing high-order approximations and applying these results to random continued fractions.

Contribution

It introduces a $k^{ ext{th}}$-order approximation method for invariant densities in perturbed random dynamical systems with spectral gaps, specifically applied to random continued fractions.

Findings

Derived a $k^{ ext{th}}$-order approximation formula for invariant densities.

Applied the approximation to random continued fractions.

Demonstrated the effectiveness of the method in specific examples.

Abstract

We study perturbations of random dynamical systems whose associated transfer operators admit a uniform spectral gap. We provide a $k^{\text{th}}$-order approximation for the invariant density of the associated random dynamical system. We apply our result to random continued fractions.