A pseudospectral approach to rigorous numerical estimation of resonances of transfer operators

Alex Blumenthal; Isaia Nisoli; Toby Taylor-Crush

REVIEW 2 major objections 4 minor 1 cited by

A pseudospectral approach to rigorous numerical estimation of resonances of transfer operators

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Pith's one-line read Certified pseudospectral enclosures rigorously locate transfer-operator resonances.

desk verdict A genuinely new certified enclosure/exclosure method for transfer-operator resonances, with working code and a clean benchmark, but the multiplicity-preservation proof does not hold as written because the Keller–Liverani hypothesis fails. read the letter →

arxiv 2507.09021 v2 pith:EGNHKK45 submitted 2025-07-11 math.DS math.FA

Alex Blumenthal , Isaia Nisoli , Toby Taylor-Crush This is my paper · ORCID

classification math.DSmath.FA MSC 37C3047A1065G99

keywords Ruelle-Pollicottresonancestransferoperatorspseudospectrumrigorousnumericscomputer-assistedproofspectralapproximationuniformlyexpandingmapsvalidated

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Resonances of chaotic dynamical systems are isolated eigenvalues of transfer operators, and proving where they lie for a specific map has been hard because any finite computation must tame an infinite-dimensional tail. This paper establishes that a classical finite-dimensional inclusion principle—the spectrum of an operator lies inside the pseudospectrum of any sufficiently close approximation—can be made to work for transfer operators, provided one measures the approximation error in a mixed norm and controls the eigenfunction norm ratio with a Doeblin–Fortet–Lasota–Yorke inequality. The method then delivers certified enclosures and, just as importantly, certified exclusions: for the perturbed doubling map $T(z)=iz^2\exp((1/2-b\pi)(z-z^{-1}))$ with $b=5/64+1/128+1/256$, the full spectrum is contained in the union of four disks, with exactly one simple resonance in each of three of them and no spectrum outside the union. A Blaschke-product example with exactly known spectrum confirms the framework reproduces the true answer.

What carries the argument

The engine is a mixed-norm version of the Householder spectral-inclusion bound: for $Af=\lambda f$, one gets $|f|_w \le \|(\lambda-\tilde A)^{-1}\|_w \,\|A-\tilde A\|_{s\to w}\,|f|_s$, so $\lambda$ enters the $\delta$-pseudospectrum of $\tilde A$ once $\delta \ge \gamma_A(\lambda)\|A-\tilde A\|_{s\to w}$. In the analytic setting, $\tilde A=L_{T,K}$ is the Galerkin operator, the mixed error decays like $e^{-2\pi K(\alpha-\eta)}$ up to a constant, and $\gamma_A(\lambda)$ is bounded via the interpolation inequality between $A_0$, $A_\eta$, and $A_\alpha$. Multiplicity preservation comes from the spectral stability theorem for DFLY operators applied along the straight-line homotopy $A_t=tL_T+(1-t)L_{T,K}$, while subharmonicity of the resolvent norm lets the pseudospectral check be performed on one-dimensional contour curves rather than open regions of the plane. The finite-dimensional certification then combines an approximate Schur decomposition with ball-matrix error bounds and verified bounds on singular values of the resolvent matrices.

What would settle it

For the perturbed doubling map, recompute the certified resolvent norm $r_{L_{T,K}}(z)$ in the $A_0$ norm at a dense mesh on the four circle boundaries using an independent validated linear algebra package; if any value strictly exceeds $\delta^{-1}=1.04\times10^{19}$ at a point outside the four disks, or if the certified Schur eigenvalues of $L_{T,K}$ do not all lie in the disks, Theorem 4.8 fails. Running the published scripts with a different but still validated interval-arithmetic implementation should reproduce the certified maxima in the last column of Table 2.

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Extended reading notes

Core claim

The central discovery is that the Householder spectral-inclusion argument survives the infinite-dimensional setting of transfer operators if the approximation error is measured in a mixed norm and the eigenfunction 'concentration' ratio $\gamma_A(\lambda)$ is controlled a priori. For an analytic uniformly expanding circle map satisfying Assumption 3.6, the Galerkin truncation $L_{T,K}=\Pi_K L_T \Pi_K$ converges in the mixed norm $A_\alpha \to A_0$ with geometrically decaying error, and for eigenvalues away from zero the ratio $\|f\|_{A_\alpha}/\|f\|_{A_0}$ is bounded through an interpolation inequality between the Wiener algebra $A_0$ (functions with absolutely summable Fourier coefficients) and its exponentially weighted relatives. Hence whenever the $\delta$-pseudospectrum of the finite matrix $L_{T,K}$ is contained in disjoint disks, the true spectrum of $L_T$ lies in those same disks and the algebraic multiplicities agree. The paper implements this for a 128-mode truncation using validated ball arithmetic, an interval-arithmetic FFT with explicit aliasing bounds, and certified singular-value bounds, proving for the perturbed doubling map that the spectrum is confined to four disks with exactly one simple resonance in each of the three small disks.

Load-bearing premise

The certificates trust the correctness of the validated numerics stack—ball-matrix arithmetic with consistent rounding, interval-arithmetic FFT with explicit aliasing bounds, and the validated Schur and singular-value routines—so that the certified resolvent-norm upper bounds on the contour curves are genuine upper bounds.

Editorial extensions

If this is right

For the perturbed doubling map of Theorem 4.8, the transfer operator's spectrum lies inside the four certified disks, with exactly one resonance of multiplicity one in each of the three small disks and no spectrum anywhere else.
The abstract Proposition 3.14 turns certified Galerkin pseudospectra into rigorous enclosures and multiplicities for any analytic uniformly expanding circle map satisfying Assumption 3.6.
Spectral exclosures—ruling out resonances in a region—are certified directly rather than inferred from convergence of approximate eigenvalues, so the proof is immune to spectral pollution.
Because algebraic multiplicities of the finite Galerkin matrix match the true operator inside each disk, a numerical eigenvalue of $L_{T,K}$ in a small disk is a certified count of a true resonance and can serve as a starting point for local refinement.
The Blaschke benchmark confirms the framework: the certified disks reproduce the exactly known spectrum $\{0,1\}\cup\{\mu^n:n\ge1\}\cup\{\bar\mu^n:n\ge1\}$ for the degree-two Blaschke product with $\mu=\sqrt[3]{2}\,e^{i\pi/8}/8$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

By the same logic, the certified enclosure-plus-exclosure scheme should be realizable for Anosov diffeomorphisms, Axiom A flows, and dispersing billiards whenever one can produce a computable DFLY inequality and a projection with controlled mixed error; the paper's outlook points this way, but the full workflow is only demonstrated on circle maps.
The certified numbers depend on the trusted behavior of the validated ball-arithmetic and linear-algebra stack; an independent reimplementation of the Schur and singular-value certification in a separate language would provide a strong cross-check of Theorem 4.8's disk enclosures.
A natural stress test would be maps with resonances close to the essential spectrum or with tiny spectral gaps, since the adaptive contour-meshing cost is roughly proportional to the integral of the resolvent norm along the isolating curves and would reveal where the method's practical limits lie.
The pseudospectral computation converts between $\ell^1$- and $\ell^2$-type norms with a $\sqrt{n}$ factor; direct certified $\ell^1$ resolvent bounds could plausibly tighten the certified maxima and yield smaller disks at the same truncation size.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The short version: this paper has a genuinely new idea — a Householder-style pseudospectral argument that gives simultaneous rigorous enclosure and exclosure for Ruelle–Pollicott resonances of transfer operators, with multiplicity preservation claimed. That last part is the weak point as written.

What’s actually new: adapting the finite-dimensional pseudospectral inclusion theorem to the DFLY/strong-weak Banach space setting, and making it computationally real for analytic expanding circle maps. Lemma 2.9 (checking resolvent bounds only on curves, via subharmonicity) is clean and makes exclosure tractable. The code and datasets are public, and the Blaschke benchmark reproduces the known spectrum [BJS17], which is a good sanity check. The perturbed doubling map result (Theorem 4.8(a)) gives certified enclosure and exclosure for that operator, and the circles F_1–F_3 are new.

Soft spots, in increasing order:

(1) Equation (12) is wrong: the dual pairing on the unit circle carries an extra z^{-1}. As written, the contour-shift bound in Lemma 3.7 is not derived. The statement survives with the correct formula, so this is a fixable typo, but it is in a central lemma.

(2) The multiplicity claims (Theorem 4.8(b), and the analog for the Blaschke example) rely on the path argument in the proof sketch of Proposition 3.14. That argument invokes Keller–Liverani’s Lemma 2.12, which requires the outer exclusion radius μ to satisfy μ > β. In the perturbed doubling application β ≈ 70.7 and μ = 0.21; the hypothesis is not met. Also, the claimed DFLY inequality with C_2 = 0 is not established by Lemma 3.7, which only gives a one-step bound between different spaces, not a contraction on the strong space. The enclosure half does not depend on this, but the simplicity and multiplicity assertions in the abstract and Theorem 1.1 are not proven as written.

(3) The certificates depend on BallArithmetic.jl and a particular OpenBLAS rounding mode, with code and logs but no independent proof of the library’s correctness. That is normal in computer-assisted proofs, but it means the exclosures are as trustworthy as the toolchain.

Overall: this is a serious contribution to rigorous computational dynamics, and the enclosure/exclosure framework deserves attention. But the multiplicity result needs a repair — either a direct Riesz projection argument using compactness of L_T on A_α, or a proper small-β DFLY inequality. I’d send it to a strong referee and expect revision before acceptance.

Recommendation: engage with it. Tell the authors to fix equation (12) and to either repair the Keller–Liverani application or supply a different proof of (20).

Referee Report

2 major / 4 minor

Summary. The paper develops a computer-assisted framework for rigorous enclosure and exclosure of spectra of transfer operators, based on a Householder-type pseudospectral inclusion theorem adapted to operators satisfying a DFLY inequality. Section 2 presents the abstract theory, including multiplicity preservation via a Keller-Liverani spectral continuity argument. Section 3 specializes to analytic expanding maps of the circle, using the Wiener algebras A_eta as a Banach scale and Galerkin truncation L_{T,K}; Proposition 3.14 is the main bridge from certified pseudospectral enclosures of the finite matrix to the true spectrum. Section 4 reports two applications: a Blaschke product whose spectrum is known exactly (benchmark), and a perturbed doubling map for which the paper claims a new certified containment and one simple resonance in each of three small disks (Theorem 4.8). Section 5 explains the validated numerical tools: approximate Schur decomposition, Rump's approximate SVD, ball-matrix arithmetic, and a validated FFT. The paper is accompanied by public code and datasets.

Significance. The methodological idea is attractive and the computational implementation is exemplary in its transparency: the authors provide code, certification logs, and datasets, and the Blaschke example gives an external benchmark that the computed enclosure indeed matches the known spectrum from [BJS17]. If the proof gaps I identify are repaired, the approach would be a valuable new tool for certified spectral computation of transfer operators, particularly for the exclosure problem, which is harder than enclosure and rarely addressed rigorously. However, the main results currently rest on two load-bearing proof steps: the Fourier coefficient formula in Lemma 3.7 and the Keller-Liverani spectral continuity invocation in Proposition 3.14, neither of which is valid as written. These must be corrected or replaced before the claims can be accepted.

major comments (2)

[Lemma 3.7, Eq. (12)] The stated change-of-variables formula c_k(L_T phi) = (1/2 pi i) int_{S^1} phi(z) T(z)^{-k-1} dz is not correct for the transfer operator of Definition 3.5. With normalized Lebesgue measure, the defining duality is int (L_T f)(x) g(x) dtheta(x) = int f(y) g(T(y)) dtheta(y), which gives c_k(L_T phi) = (1/2 pi i) int phi(z) T(z)^{-k} z^{-1} dz. The printed formula has an extra factor z/T(z); for the doubling map T(z)=z^2 and phi(z)=z^2, one has L_T phi = z and c_1(L_T phi)=1, whereas Eq. (12) evaluates to (1/2 pi i) int z^{-2} dz = 0. Since the contour-shift bound (11) is derived from Eq. (12), the proof of Lemma 3.7 is invalid as written. Please correct the formula and verify that the estimates (11), (18), and Lemma 3.13 survive the correction.
[Proposition 3.14, proof sketch (I)-(III); Lemma 2.12] The proof of (20) invokes Lemma 2.12 for the path A_t = t L_T + (1-t) L_{T,K} with strong-weak pair (A_alpha, A_eta), claiming a DFLY inequality with C_1=1, C_2=0 and beta = ||L_T||_{A_eta -> A_alpha}. Lemma 2.12 requires mu in (beta, M), where mu is the radius of the outer exclusion circle. In Theorem 4.8 this radius is r = 0.21 (Table 2, F_0), while for the stated parameters (eta = 0.22211055, rho = 0.312891, alpha = 0.308389) Lemma 3.7 gives beta >= 1 + 2/(e^{2 pi (rho-alpha)} - 1) ~ 70.7. Hence mu = 0.21 < beta, so the hypothesis of Lemma 2.12 is violated and the spectral-continuity argument does not go through. Consequently the multiplicity claims m_{F_i}(L_T) = 1 in Theorem 4.8(b), and similarly in Theorem 4.6(b), are not established by the given proof. A direct Riesz-projection argument, or a Lasota-Yorke inequality with beta < mu, is required; this is a load-bearing gap, not a matter of presentation.

minor comments (4)

[Eq. (18)] Equation (18) uses the symbol r both for the fixed outer radius and for the bound being defined; rename the outer radius (e.g., \bar r) to remove the collision.
[Proof comments for Theorem 4.8] In the proof comments for Theorem 4.8, 'Additional details are as in the proof of Theorem 4.3' should presumably refer to Theorem 4.6.
[Figure 4] The caption of Figure 4 mislabels the content: the first line says 'Circle enclosure for the Blaschke product,' but the figure displays the perturbed doubling map.
[Section 5.3] Section 5.3 should state the exact versions of BallArithmetic.jl, OpenBLAS, and the Julia packages used for the certificates and link the test/CI logs, so that the consistent-rounding claim can be independently reproduced.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the spectral enclosures are certified a posteriori from validated matrix computations; only minor self-citations to the authors' numerical tooling appear.

full rationale

The derivation is self-contained against widely-used, external tools: Theorem 1.2 and Theorem 1.3 are classical Householder and pseudospectrum results, Lemma 2.12 is the external Keller-Liverani stability theorem, and the Blaschke benchmark in Theorem 4.6 reproduces the independent spectrum of [BJS17]. Proposition 3.14 is a genuine transfer statement: it assumes a certified mixed-norm approximation error and certified resolvent bounds on the contours, and concludes spectral enclosure plus multiplicity equality. The disk centers and radii in Tables 1 and 2 are derived from nonrigorous Schur and SVD computations, but the certificates do not assume the existence or multiplicity of the target resonances; they certify a delta-pseudospectral enclosure for the finite matrix L_{T,K}, which is a different object from the claimed spectrum of L_T. This is a posteriori certification, not fitting an input that already contains the answer. The perturbed doubling map result is new and is not obtained by construction from any input. Self-citations appear at [NTC23] (validated FFT), [FN23] (ball arithmetic), and [BNTC25a,b] (datasets); these are tooling references, load-bearing for the rigorous numerics but not for the spectral conclusion, and they are code-reproduced and publicly available. One concern identified in review is not circularity: the proof sketch of Proposition 3.14 invokes Lemma 2.12 with outer radius mu = 0.21, while the DFLY constant beta from Lemma 3.7 is approximately 70.7, so the stated constants do not satisfy the lemma's hypothesis mu in (beta, M); this is a correctness or proof-gap issue, not an identity between input and output. Accordingly the circularity score is 1, reflecting minor self-citation without any reduction of the central claim to its inputs.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new physical or mathematical entities. Its degrees of freedom are computational parameters (space weights, truncation order, disk geometry), all certified a posteriori. The proofs import standard results (DFLY/Hennion, Keller-Liverani, Rump, subharmonicity) and rely on a trusted validated-numerics toolchain.

free parameters (6)

eta (Banach space weight) = Blaschke: 0.49149149; doubling: 0.22211055
Chosen by nonrigorous search, then certified via interval arithmetic to satisfy Assumption 3.6 (Lemma 4.5/4.7).
rho (outer annulus parameter) = Blaschke: 0.583052; doubling: 0.312891
Certified lower bound on the expansion scale; used in all norm estimates.
alpha (intermediate space weight) = Blaschke: 0.5758488557738615; doubling: 0.308389
Hand-chosen between eta and rho to balance the eigenfunction ratio bound (18) and the discretization error (Lemma 3.13).
K (Galerkin truncation order) = 128
Chosen so that the certified discretization error is small enough for the pseudospectral level delta required in both examples.
r (radius of central disk F_0) = Blaschke: 0.51; doubling: 0.21
Hand-chosen to separate the essential-spectral block around 0 from the isolated resonances; the exclosure is then certified.
Disks F_2/F_3 centers and radii = Blaschke: 0.4899611+i0.2029485, r=0.01; doubling: -0.0528637+i0.2062498, r=0.001
Centers taken from nonrigorous Schur decomposition; radii chosen small; the certification then proves the true spectrum lies inside and absent outside.

assumptions (6)

domain assumption Assumption 3.6: T admits an analytic extension to U_{e^{2 pi eta} + epsilon} and U_{e^{2 pi rho}} intersects T(partial U_{e^{2 pi eta}}) trivially.
Verified for each example by certified interval arithmetic (Lemmas 4.5 and 4.7), not by a closed-form proof.
standard math Keller-Liverani spectral stability Lemma 2.12 ([KL99, Cor 1])
Used to prove multiplicity preservation along the operator path in Theorem 2.8 and Proposition 3.14.
standard math Hennion-Nussbaum description of essential spectrum (Proposition 2.5)
Gives isolated point spectrum outside B_beta(0) under the DFLY inequality.
standard math Subharmonicity of the resolvent norm and Globevnik's non-constancy result ([Glo76, Prop 1]) for Lemma 2.9
Reduces pseudospectrum containment check from a 2D region to a finite union of curves; the paper notes Shargorodsky's caution in Remark 2.13.
standard math Rump's verified SVD bounds (Proposition 5.5, [Rum11, Thm 3.1])
Provides certified estimates for the smallest singular value, hence resolvent norms of the approximate matrices.
domain assumption Correct behavior of BallArithmetic.jl, IntervalArithmetic.jl and consistently-rounded OpenBLAS (ConsistentFPCSR=1)
Load-bearing for all numerical certificates; the text provides code and logs but no proof of the libraries themselves.

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Pith. "Pith review of A pseudospectral approach to rigorous numerical estimation of resonances of transfer operators." pith.science (2026). https://pith.science/paper/EGNHKK45

@misc{pith2026250709021,
  author       = {Pith},
  title        = {Pith review of: A pseudospectral approach to rigorous numerical estimation of resonances of transfer operators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EGNHKK45}},
  note         = {Machine review of arXiv:2507.09021}
}

read the original abstract

{Ruelle-Pollicott} resonances, isolated eigenvalues of a transfer operator acting on suitably chosen Banach spaces, play a fundamental role in understanding the statistical properties of chaotic dynamical systems. In this paper, we introduce a pseudospectral approach, inspired by Householder's theorem, for the rigorous, computer-assisted estimation of resonances, providing regions where resonances must exist and precluding the presence of resonances elsewhere. The approach is general, and applies to the transfer operators of a wide variety of chaotic systems, including Anosov/ Axiom A diffeomorphisms and piecewise expanding maps. We implement this approach computationally for a class of analytic uniformly expanding maps of the circle. We anticipate that the pseudospectral framework developed here will be broadly applicable to other spectral problems in dynamical systems and beyond.

Figures

Figures reproduced from arXiv: 2507.09021 by the authors.

**Figure 1.** Circle enclosure for the perturbed doubling map (Theorem 1.1). Figure 1a depicts the circles F0, F1 in Theorem 1.1 enclosing 0 and 1, respectively. Figure 1b is zoomedin to depict the circle F2 enclosing resonance λ, which is small and situated close to F0. Circle F3 enclosing resonance λ¯ is similar. In this paper, we put forward an approach for the rigorous, computerassisted estimation of spectrum of a wide cla… view at source ↗

**Figure 2.** Numerical plot (non-certified) of the pseudospectrum levels with respect to the spectral norm of a perturbation A˜ of the diagonal matrix A with entries (1.0, 1.2, 2.0, 3.0), such that ||A − A˜||∞ ≤ 0.0001. The contour curves correspond to levels of − log10(||(z−A˜) −1 ||2). We refer to [TE05] for the numerical methods used to compute this graph. Note how the pseudospectrum behaves between 1.0 and 1.2. Theorem 1.3… view at source ↗

**Figure 3.** Circle enclosure for the Blaschke product (a) Non validated ℓ 2 -pseudospectra of the perturbed doubling map product (23) (b) Computed enclosures, refer to [PITH_FULL_IMAGE:figures/full_fig_p030_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Circle enclosure for the Blaschke product Lemma 4.7. The map T as in (23) satisfies Assumption 3.6 with ρ = 0.312891 and η = 0.22211055. Theorem 4.8. Let T be as in (23). Then, (a) σ(LT ) ⊂ ∪3 0Fi; and (b) mFi (LT ) = 1 for i = 1, 2, 3; where F0, F1, F2, F3 are the ope…

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Works this paper leans on

17 extracted references · 10 canonical work pages · cited by 1 Pith paper

[13]

Expanding maps of the circle rerevisited: positive Lyapunov exponents in a rich family

PSEUDOSPECTRAL APPROACH TO RESONANCES 47 [PRS06] Enrique R Pujals, Leonel Robert, and Michael Shub. Expanding maps of the circle rerevisited: positive Lyapunov exponents in a rich family. Ergodic Theory and Dynamical Systems, 26(6):1931–1937,

work page 1931
[15]

[SB21] Ay¸ se G¨ uven Sarıhan and Oscar F Bandtlow

URL:https://github.com/JuliaIntervals/IntervalArithmetic.jl, doi:10.5281/zenodo.3336308. [SB21] Ay¸ se G¨ uven Sarıhan and Oscar F Bandtlow. Quantitative spectral per- turbation theory for compact operators on a hilbert space.Linear Algebra and its Applications, 610:169–202,

work page doi:10.5281/zenodo.3336308
[192]

[Nus70] Roger D Nussbaum

doi:10.1007/s10955-023-03174-8. [Nus70] Roger D Nussbaum. The radius of the essential spectrum.Duke Mathe- matical Journal, 37(3):473–478,

work page doi:10.1007/s10955-023-03174-8
[1976]

Anosov flows and dynamical zeta functions.Ann

[GLP13] Paolo Giulietti, Carlangelo Liverani, and Mark Pollicott. Anosov flows and dynamical zeta functions.Ann. Math. (2), 178(2):687–773, 2013.doi: 10.4007/annals.2013.178.2.6. [GMN20] Stefano Galatolo, Maurizio Monge, and Isaia Nisoli. Existence of noise induced order, a computer aided proof.Nonlinearity, 33(9):4237,

work page doi:10.4007/annals.2013.178.2.6 2013
[2000]

The quest for the ultimate anisotropic banach space

[Bal16] Viviane Baladi. The quest for the ultimate anisotropic banach space. Journal of Statistical Physics, 166(3–4):525–557, November 2016.doi: 10.1007/s10955-016-1663-0. [Bal18] Viviane Baladi.Dynamical zeta functions and dynamical determinants for hyperbolic maps. Springer,

work page doi:10.1007/s10955-016-1663-0 2016
[2003]

Banach spaces adapted to Anosov systems.Ergodic Theory Dyn

PSEUDOSPECTRAL APPROACH TO RESONANCES 45 [GL06] S´ ebastien Gou¨ ezel and Carlangelo Liverani. Banach spaces adapted to Anosov systems.Ergodic Theory Dyn. Syst., 26(1):189–217, 2006.doi: 10.1017/S0143385705000374. [Glo76] Josip Globevnik. Norm-constant analytic functions and equivalent norms. Illinois Journal of Mathematics, 20(3):503–506,

doi:10.1017/s0143385705000374 2006
[2006]

[Rum99] Siegfried M. Rump. Fast and parallel interval arithmetic.BIT Numerical Mathematics, 39:534–554, 1999.doi:10.1023/A:1022374804152. [Rum11] S. M. Rump. Verified bounds for singular values, in particular for the spectral norm of a matrix and its inverse.BIT Numerical Mathematics, 51:367–384,

work page doi:10.1023/a:1022374804152 1999
[2011]

Colbrook, Oliver Junge, P´ eter Koltai, and Ju- lia Slipantschuk

[HCJ+25] April Herwig, Matthew J. Colbrook, Oliver Junge, P´ eter Koltai, and Ju- lia Slipantschuk. Avoiding spectral pollution for transfer operators using residuals, 2025.arXiv:2507.16915. [Hen62] Peter Henrici. Bounds for iterates, inverses, spectral variation and fields of values of non-normal matrices.Numerische Mathematik, 4:24–40,

arXiv 2025

Show all 17 references

[2012]

Eigenvalue enclo- sure logs for Blaschke product example, 2025.doi:10.7910/DVN/QFAIJL

[BNTC25a] Alex Blumenthal, Isaia Nisoli, and Toby Taylor-Crush. Eigenvalue enclo- sure logs for Blaschke product example, 2025.doi:10.7910/DVN/QFAIJL. [BNTC25b] Alex Blumenthal, Isaia Nisoli, and Toby Taylor-Crush. Eigenvalue enclo- sure logs for perturbed doubling map example...

2025 doi
[2014]

Lagrange approximation of transfer operators associated with holomorphic data.arXiv preprint arXiv:2004.03534,

[BS20] Oscar F Bandtlow and Julia Slipantschuk. Lagrange approximation of transfer operators associated with holomorphic data.arXiv preprint arXiv:2004.03534,

2004 arXiv
[2015]

On the approximation of operators and the convergence of the spectra of the approximants

[Wol98] MPH Wolff. On the approximation of operators and the convergence of the spectra of the approximants. InRecent Progress in Operator Theory: International Workshop on Operator Theory and Applications, IWOTA 95, in Regensburg, July 31–August 4, 1995, pages 279–283. Springer,

1995
[2017]

Resonances for rational Anosov maps on the torus.arXiv preprint arXiv:2211.05925,

[SBJ22] Julia Slipantschuk, Oscar F Bandtlow, and Wolfram Just. Resonances for rational Anosov maps on the torus.arXiv preprint arXiv:2211.05925,

arXiv
[2019]

[CT24a] Matthew J

doi:10.1103/PhysRevLett.122.250201. [CT24a] Matthew J. Colbrook and Alex Townsend. Rigorous data-driven compu- tation of spectral properties of koopman operators for dynamical systems. Communications on Pure and Applied Mathematics,

doi
[2021]

Demers and Carlangelo Liverani

[DL08] Mark F. Demers and Carlangelo Liverani. Stability of statistical prop- erties in two-dimensional piecewise hyperbolic maps.Trans. Am. Math. Soc., 360(9):4777–4814, 2008.doi:10.1090/S0002-9947-08-04464-4. [dlLMJ16] Rafael de la Llave and Jason D Mireles James. Connecting...

2008 doi
[2022]

Colbrook, and Gustav Conradie

[BCC25] Nicolas Boull´ e, Matthew J. Colbrook, and Gustav Conradie. Convergent methods for Koopman operators on reproducing kernel Hilbert spaces, 2025.arXiv:2506.15782. PSEUDOSPECTRAL APPROACH TO RESONANCES 43 [BDL18] Viviane Baladi, Mark F Demers, and Carlangelo Liverani. Ex...

2025 arXiv
[2024]

Colbrook and Alex Townsend

44 ALEX BLUMENTHAL, ISAIA NISOLI, AND TOBY TAYLOR-CRUSH [CT24b] Matthew J. Colbrook and Alex Townsend. Rigorous data-driven computa- tion of spectral properties of Koopman operators for dynamical systems. Comm. Pure Appl. Math., 77:221–283, 2024.doi:10.1002/cpa.22125. [DF37] W...

2024 doi
[2026]

Blum and O

[BA00] G. Blum and O. Agam. Leading ruelle resonances of chaotic maps.Phys. Rev. E, 62:1977–82,

1977

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [13]

Expanding maps of the circle rerevisited: positive Lyapunov exponents in a rich family

PSEUDOSPECTRAL APPROACH TO RESONANCES 47 [PRS06] Enrique R Pujals, Leonel Robert, and Michael Shub. Expanding maps of the circle rerevisited: positive Lyapunov exponents in a rich family. Ergodic Theory and Dynamical Systems, 26(6):1931–1937,

work page 1931

[2] [15]

[SB21] Ay¸ se G¨ uven Sarıhan and Oscar F Bandtlow

URL:https://github.com/JuliaIntervals/IntervalArithmetic.jl, doi:10.5281/zenodo.3336308. [SB21] Ay¸ se G¨ uven Sarıhan and Oscar F Bandtlow. Quantitative spectral per- turbation theory for compact operators on a hilbert space.Linear Algebra and its Applications, 610:169–202,

work page doi:10.5281/zenodo.3336308

[3] [192]

[Nus70] Roger D Nussbaum

doi:10.1007/s10955-023-03174-8. [Nus70] Roger D Nussbaum. The radius of the essential spectrum.Duke Mathe- matical Journal, 37(3):473–478,

work page doi:10.1007/s10955-023-03174-8

[4] [1976]

Anosov flows and dynamical zeta functions.Ann

[GLP13] Paolo Giulietti, Carlangelo Liverani, and Mark Pollicott. Anosov flows and dynamical zeta functions.Ann. Math. (2), 178(2):687–773, 2013.doi: 10.4007/annals.2013.178.2.6. [GMN20] Stefano Galatolo, Maurizio Monge, and Isaia Nisoli. Existence of noise induced order, a computer aided proof.Nonlinearity, 33(9):4237,

work page doi:10.4007/annals.2013.178.2.6 2013

[5] [2000]

The quest for the ultimate anisotropic banach space

[Bal16] Viviane Baladi. The quest for the ultimate anisotropic banach space. Journal of Statistical Physics, 166(3–4):525–557, November 2016.doi: 10.1007/s10955-016-1663-0. [Bal18] Viviane Baladi.Dynamical zeta functions and dynamical determinants for hyperbolic maps. Springer,

work page doi:10.1007/s10955-016-1663-0 2016

[6] [2003]

Banach spaces adapted to Anosov systems.Ergodic Theory Dyn

PSEUDOSPECTRAL APPROACH TO RESONANCES 45 [GL06] S´ ebastien Gou¨ ezel and Carlangelo Liverani. Banach spaces adapted to Anosov systems.Ergodic Theory Dyn. Syst., 26(1):189–217, 2006.doi: 10.1017/S0143385705000374. [Glo76] Josip Globevnik. Norm-constant analytic functions and equivalent norms. Illinois Journal of Mathematics, 20(3):503–506,

doi:10.1017/s0143385705000374 2006

[7] [2006]

[Rum99] Siegfried M. Rump. Fast and parallel interval arithmetic.BIT Numerical Mathematics, 39:534–554, 1999.doi:10.1023/A:1022374804152. [Rum11] S. M. Rump. Verified bounds for singular values, in particular for the spectral norm of a matrix and its inverse.BIT Numerical Mathematics, 51:367–384,

work page doi:10.1023/a:1022374804152 1999

[8] [2011]

Colbrook, Oliver Junge, P´ eter Koltai, and Ju- lia Slipantschuk

[HCJ+25] April Herwig, Matthew J. Colbrook, Oliver Junge, P´ eter Koltai, and Ju- lia Slipantschuk. Avoiding spectral pollution for transfer operators using residuals, 2025.arXiv:2507.16915. [Hen62] Peter Henrici. Bounds for iterates, inverses, spectral variation and fields of values of non-normal matrices.Numerische Mathematik, 4:24–40,

arXiv 2025