Effective estimates of ergodic quantities illustrated on the Bolyai-R\'enyi map

TL;DR

This paper introduces a practical method for high-precision estimation of ergodic quantities related to transfer operators, demonstrated on the Bolyai-Rényi map, combining approximation bounds with a min-max approach.

Contribution

It presents a novel combination of explicit error bounds and min-max methods for rigorous high-precision estimates of ergodic quantities.

Findings

Improved rigorous estimates for ergodic quantities of the Bolyai-Rényi map.

Demonstrated effectiveness of the method in high-precision calculations.

Applicable to a range of transfer operator-related ergodic problems.

Abstract

We present a practical and effective method for rigorously estimating quantities associated to top eigenvalues of transfer operators to very high precision. The method combines explicit error bounds of the Lagrange-Chebyshev approximation with an established min-max method. We illustrate its applicability by significantly improving rigorous estimates on various ergodic quantities associated to the Bolyai-R\'enyi map.