Invariant densities and escape rates: Rigorous and computable approximations in the $L^{\infty}$-norm

TL;DR

This paper develops rigorous, computable methods for approximating invariant densities and escape rates of interval maps using piecewise linear discretizations, with proven convergence in the supremum norm.

Contribution

It introduces a novel approach for pointwise approximation of invariant densities with proven convergence rates in the $L^{ abla}$-norm, complementing existing $L^1$ methods.

Findings

The methods provide rigorous pointwise invariant density approximations.

Convergence order in the $L^{ abla}$-norm is established.

Computational examples demonstrate the scheme's effectiveness.

Abstract

In this article we study a piecewise linear discretization schemes for transfer operators (Perron-Frobenius operators) associated with interval maps. We show how these can be used to provide rigorous {\bf pointwise} approximations for invariant densities of Markov interval maps. We also derive the order of convergence of the approximate invariant density to the real one in the $L^{\infty}$-norm. The outcome of this paper complements rigorous results on $L^1$ approximations of invariant densities \cite{KMY} and recent results on the formulae of escape rates of open dynamical systems \cite{KL2}. We implement our computations on two examples (one rigorous and one non-rigorous) to illustrate the feasibility and efficiency of our schemes.