arXiv:1806.10884v3

Riesz products and spectral decompositions for rank 1 measure preserving transformations

Yury A. Neretin

math.DSmath.FAmath.SP47A3528D0537A4042A5547B15

Paper metadata

Original submission: June 28, 2018
Latest revision: July 26, 2026 · version 3
Claimed MSC: 47A35, 28D05, 37A40, 42A55, 47B15
arXiv categories: math.DS, math.FA, math.SP
Paper license: arXiv non-exclusive distribution license ↗
Comments: 24 pages, extended version

Abstract

We consider rank one measure preserving transformations $g$ and the corresponding Koopman unitary operators $U(g)$ . It is known that a generic (in the sense of Baire category) measure preserving transformation has rank one, spectral measure of $U(g)$ is purely singular and is given by a Riesz product. For such transformations we write explicitly spectral decompositions of operators $U(g)$ in generalized eigenfunctions.

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Audited against arXiv v3

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

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Generated August 18, 2026

01Statements2 reported findingsCorrect

The two explicit spectral assertions are supported. Theorem 2.1 constructs a unitary model in which the rank-one Koopman operator is multiplication by the circle coordinate against the generalized Riesz product $\kappa$ . Proposition 4.1 then gives the printed orthogonal rational-function basis, including its norm formula.

Theorem 2.1Correct

Unitary Riesz-product model for the rank-one Koopman operator

Pages 3–7 · Theorem 2.1 · arXiv:1806.10884v3

Assuming the polynomials $\Theta_k$ have no zeros on the unit circle, the map $R$ defined on cylinder indicators extends from $L^2(V)$ to a unitary map onto $L^2(S^1,\kappa)$ , where $\kappa$ is the displayed generalized Riesz product. Under this map the Koopman operator $U$ becomes multiplication by $z$ . The no-zero hypothesis is exactly what makes the rational functions defining $R$ well defined; the conclusion does not silently assert the result for the excluded singular case.

Proposition 4.1Correct

Explicit orthogonal basis in the Riesz-product space

Pages 19–20 · Proposition 4.1 · arXiv:1806.10884v3

The functions $\Phi[k,j,r;l,N]$ range over a disjoint parametrization of the characters on the compact residue-ring components of $V$ . Applying the unitary map $R$ from Theorem 2.1 carries that character basis to the displayed rational functions. Unitarity preserves orthogonality and gives the stated squared norm from the measure of the corresponding ball, so the proposition supplies a complete basis rather than merely an orthogonal family.

02Proofs2 reported findingsCorrect

Coefficient identities, isometry, and surjectivity. Lemmas 3.1–3.5 compute the tower-coordinate exponent $\Upsilon$ and show that the finite Riesz products have the Fourier coefficients required by the intersections of cylinder balls. Lemma 3.6 therefore proves preservation of inner products first for equal-level indicators and then for all locally constant functions. Lemma 3.7 verifies $R(Uf)=zR(f)$ on that dense domain. Since $R$ sends the base indicator and all of its $U$ -iterates to the Laurent monomials, its isometric range contains a dense set in $L^2(S^1,\kappa)$ , which proves surjectivity rather than only an isometric embedding.

Intertwining constructionCorrect and complete

Coefficient identities, isometry, and surjectivity

Pages 7–14 · proof of Theorem 2.1 · arXiv:1806.10884v3

Lemmas 3.1–3.5 compute the tower-coordinate exponent $\Upsilon$ and show that the finite Riesz products have the Fourier coefficients required by the intersections of cylinder balls. Lemma 3.6 therefore proves preservation of inner products first for equal-level indicators and then for all locally constant functions. Lemma 3.7 verifies $R(Uf)=zR(f)$ on that dense domain. Since $R$ sends the base indicator and all of its $U$ -iterates to the Laurent monomials, its isometric range contains a dense set in $L^2(S^1,\kappa)$ , which proves surjectivity rather than only an isometric embedding.

Proposition 4.1Correct and complete

Character decomposition and transport of the basis

Pages 19–20 · proof of Proposition 4.1 · arXiv:1806.10884v3

The proof writes each compact residue-ring group as an inverse limit and its dual as the direct limit of the finite cyclic duals. Separating characters by the first level at which their frequency is nontrivial gives exactly the restrictions on $l$ and $N$ in the proposition. The decomposition of $V$ into the corresponding balls is disjoint and exhaustive, and transporting these character bases through $R$ proves both completeness and the norm formula.

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Paper: arXiv:1806.10884v3
Authors listed: Yury A. Neretin
Audit date: August 18, 2026

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Riesz products and spectral decompositions for rank 1 measure preserving transformations

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Unitary Riesz-product model for the rank-one Koopman operator

Explicit orthogonal basis in the Riesz-product space

Coefficient identities, isometry, and surjectivity

Character decomposition and transport of the basis

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