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Certified spectral approximation of transfer operators and the Gauss map

Isaia Nisoli

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.

Certified spectral approximation of transfer operators and the Gauss map

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 nonzero eigenvalues of the Gauss--Kuzmin--Wirsing operator to at least 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.
Paper Structure (42 sections, 38 theorems, 172 equations, 3 figures, 1 table)

This paper contains 42 sections, 38 theorems, 172 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction and overview
  2. Spectral questions as certified linear algebra.
  3. The Gauss--Kuzmin--Wirsing problem.
  4. Certified spectral data for the Gauss map.
  5. Spectral gap as an output, not an input.
  6. The general framework.
  7. Guaranteed convergence of the full spectral picture.
  8. Computability of isolated spectral data.
  9. Related approaches.
  10. Organization.
  11. A posteriori validated spectral certification: an abstract template
  12. Resolvent inequalities and the Householder argument
  13. Resolvent perturbation bound and spectral exclusions
  14. Riesz projectors, multiplicity, and invariant subspaces
  15. Coarse--fine approach
  16. ...and 27 more sections

Key Result

Theorem 1.1

The first $50$ nonzero eigenvalues of the Gauss--Kuzmin--Wirsing operator, realized as $L:=S\mathcal{L}:H^2(D_1)\to H^2(D_1)$(Section sec:gkw-hardy), are real and simple. Certified enclosures are given in the supplementary material, each with at least $90$ certified decimal digits. In particular, th Certified enclosures for the corresponding eigenvectors and Riesz spectral projectors are provided,

Figures (3)

  • Figure 1: The first six certified eigenfunctions $v_1, \ldots, v_6$ of the GKW transfer operator on $[0,1]$, computed at $K = 512$ (1024-bit precision). Each panel shows the eigenvalue $\lambda_j$ and spectral coefficient $\ell_j(\mathbf{1})$. The invariant density $v_1 = \frac{1}{\ln 2}\cdot\frac{1}{1+x}$ (top left) is positive; higher eigenfunctions develop a singularity at $x = 0$ and oscillations that reflect the Markov partition of the Gauss map.
  • Figure 2: Integrated spectral expansion $F_n^{(j_0)}(x) = \int_0^x \sum_{j \geq j_0} \lambda_j^n\,\ell_j(\mathbf{1})\,v_j(t)\,dt$ with the first $j_0 - 1$ eigenfunctions cumulatively removed. Each panel shows $n = 1$ (blue), $n = 2$ (red), $n = 5$ (green). Dotted vertical lines mark the Markov partition boundaries $x = 1/k$. For $j_0 = 6$, $n = 5$, the signal amplitude is ${\sim}\,10^{-16}$ while the $L^\infty$ error bound is ${\leq}\,10^{-105}$, confirming that the plotted curves are rigorous to plotting accuracy.
  • Figure 3: Convergence of the spectral approximant $S_{50}(n,x) = \sum_{j=1}^{50} \lambda_j^n\,\ell_j(\mathbf{1})\,v_j(x)$ (normalized by $\lambda_1^n$) to the invariant density $\ell_1(\mathbf{1})\,v_1(x) = \frac{1}{\ln 2(1+x)}$ (dashed black). The spectral gap $|\lambda_2/\lambda_1| \approx 0.304$ governs the convergence rate: by $n = 5$ the approximation is indistinguishable from the limit.

Theorems & Definitions (86)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.2: Kato1995
  • Corollary 2.3
  • proof
  • Proposition 2.4
  • proof
  • Definition 2.5
  • Proposition 2.6
  • proof
  • ...and 76 more