Certified spectral approximation of transfer operators and the Gauss map

TL;DR

This paper develops a rigorous method to approximate the entire spectral structure of transfer operators, including eigenvalues and eigenvectors, with certified error bounds, applied here to the Gauss map's transfer operator.

Contribution

It extends spectral approximation techniques to certify the full spectrum of transfer operators with explicit error bounds, including the Gauss map, without spectral pollution.

Findings

Certified the first 50 nonzero eigenvalues of the Gauss--Kuzmin--Wirsing operator to 90 decimal digits.

Provided rigorous spectral expansion for Gauss--Kuzmin distributions.

Extended Li's resolution of the Ulam conjecture to the entire discrete spectrum.

Abstract

We prove that the full spectral picture of a transfer operator (every isolated eigenvalue, eigenvector, and Riesz projector outside the essential spectral radius) can be approximated to arbitrary precision by finite-rank discretizations, with no spectral pollution. The method is a~posteriori: once a computable approximation bound is available (from compactness or a Doeblin--Fortet--Lasota--Yorke inequality, which may require hyperbolicity constants and adapted Banach spaces), the spectral gap and multiplicity are certified from computed data via a resolvent perturbation bound, with no further dynamical input. This applies both to compact operators on a single Banach space and to quasi-compact operators satisfying a Doeblin--Fortet--Lasota--Yorke inequality, extending Li's resolution of the Ulam conjecture from the invariant density to the entire discrete spectrum with certified error bounds at every finite truncation. As a benchmark, we certify the first $50$ nonzero eigenvalues of the Gauss--Kuzmin--Wirsing operator to at least $90$ rigorous decimal digits, together with their eigenvectors, Riesz projectors, and spectral gap, yielding a certified spectral expansion for Gauss--Kuzmin distributions with explicit error bounds and providing a rigorous answer to the Gauss--Babenko--Knuth problem on the spectral data of the Gauss map.