Endpoint Koopman Spectral Computation: $L^1$ Residual Bounds, $L^\infty$ Instability, and Point-Spectral SCI Calibration Families
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| Formato: | Preprint |
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2026
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| _version_ | 1866916040141701120 |
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| author | Sorg, Christopher |
| author_facet | Sorg, Christopher |
| contents | We study endpoint Koopman spectral computation from the viewpoint of the Solvability Complexity Index (SCI). Let \((\mathcal X,d)\) be a compact metric space with finite Borel measure \(ω\), and let \(\mathcal K_F\) be the Koopman operator associated with a continuous nonsingular map \(F:\mathcal X\to\mathcal X\).
First, on \(L^1(\mathcal X,ω)\), we record the endpoint residual upper-bound in the target-split form. The regularized compact fixed-\(\varepsilon\) target $R_{\mathrm{ap},\varepsilon}(\mathcal K_F)$ is separated from the closed fixed-\(\varepsilon\) target $C_{\mathrm{ap},\varepsilon}(\mathcal K_F)$ and from the exact approximate point spectrum $σ_{\mathrm{ap}}(\mathcal K_F).$ This endpoint statement uses the same point-evaluation plus fixed-quadrature information model as the \(1<p<\infty\) residual theory.
Second, we isolate two obstructions at the nonseparable endpoint \(L^\infty\). Fixed quadrature schemes do not discretize the full \(L^\infty\) unit sphere, and even inside measure-preserving Cantor homeomorphisms the map $F\mapsto σ_{\mathrm{ap}}(\mathcal K_F:L^\infty\to L^\infty)$ is maximally discontinuous in Hausdorff distance under arbitrarily small uniform perturbations of \(F\). We also show that finite-period Silver-tree block constructions cannot yield analytic hardness for the \(L^\infty\) approximate point spectrum: for a fixed non-torsion \(z_0\in\mathbb T\), the condition $z_0\inσ_{\mathrm{ap}}(\mathcal K_{F}:L^\infty\to L^\infty)$ collapses to a Borel unbounded-period condition. In addition, fixed \(L^\infty\) point-eigenvalue membership is Borel in the measure-preserving continuous class, so one fixed eigenvalue cannot encode a non-Borel tree predicate.
Third, we construct Koopman point-spectrum calibration families on the Cantor space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_12044 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Endpoint Koopman Spectral Computation: $L^1$ Residual Bounds, $L^\infty$ Instability, and Point-Spectral SCI Calibration Families Sorg, Christopher Logic Numerical Analysis Dynamical Systems Spectral Theory 47B33, 47A10, 37M25, 03D78, 68Q17, 03E15 We study endpoint Koopman spectral computation from the viewpoint of the Solvability Complexity Index (SCI). Let \((\mathcal X,d)\) be a compact metric space with finite Borel measure \(ω\), and let \(\mathcal K_F\) be the Koopman operator associated with a continuous nonsingular map \(F:\mathcal X\to\mathcal X\). First, on \(L^1(\mathcal X,ω)\), we record the endpoint residual upper-bound in the target-split form. The regularized compact fixed-\(\varepsilon\) target $R_{\mathrm{ap},\varepsilon}(\mathcal K_F)$ is separated from the closed fixed-\(\varepsilon\) target $C_{\mathrm{ap},\varepsilon}(\mathcal K_F)$ and from the exact approximate point spectrum $σ_{\mathrm{ap}}(\mathcal K_F).$ This endpoint statement uses the same point-evaluation plus fixed-quadrature information model as the \(1<p<\infty\) residual theory. Second, we isolate two obstructions at the nonseparable endpoint \(L^\infty\). Fixed quadrature schemes do not discretize the full \(L^\infty\) unit sphere, and even inside measure-preserving Cantor homeomorphisms the map $F\mapsto σ_{\mathrm{ap}}(\mathcal K_F:L^\infty\to L^\infty)$ is maximally discontinuous in Hausdorff distance under arbitrarily small uniform perturbations of \(F\). We also show that finite-period Silver-tree block constructions cannot yield analytic hardness for the \(L^\infty\) approximate point spectrum: for a fixed non-torsion \(z_0\in\mathbb T\), the condition $z_0\inσ_{\mathrm{ap}}(\mathcal K_{F}:L^\infty\to L^\infty)$ collapses to a Borel unbounded-period condition. In addition, fixed \(L^\infty\) point-eigenvalue membership is Borel in the measure-preserving continuous class, so one fixed eigenvalue cannot encode a non-Borel tree predicate. Third, we construct Koopman point-spectrum calibration families on the Cantor space. |
| title | Endpoint Koopman Spectral Computation: $L^1$ Residual Bounds, $L^\infty$ Instability, and Point-Spectral SCI Calibration Families |
| topic | Logic Numerical Analysis Dynamical Systems Spectral Theory 47B33, 47A10, 37M25, 03D78, 68Q17, 03E15 |
| url | https://arxiv.org/abs/2601.12044 |