arXiv:2606.03766v1

A Graph-Based Method for Invariant Densities of Multidimensional Continued Fractions

David Siukaev

math.DSmath.CO11J7011K5005C8137A2537A44

Paper metadata

Original submission: June 2, 2026
Claimed MSC: 11J70, 11K50, 05C81, 37A25, 37A44
arXiv categories: math.DS, math.CO
Paper license: CC BY 4.0 ↗
Comments: 39 pages, 24 figures

Abstract

We propose a novel method for computing invariant densities of certain multidimensional continued fraction algorithms. Inspired by Rauzy induction, our approach builds on the formalism of simplicial systems developed by Fougeron. We introduce a win-lose induction on a graph that is conjugate to the original algorithm, and construct its natural extension by introducing the notion of a dual graph. This method explicitly reconstructs the complete dynamics of the algorithm, yielding a partition of the invariant domain of the natural extension into pieces that map to one another. We further study the ergodic properties of the algorithms within this framework; in particular, we prove that the Modified Triangle algorithm in any dimension admits a unique ergodic measure equivalent to the Lebesgue measure.

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Audited against arXiv v1

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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Detailed mathematical audit

Generated August 18, 2026

01Statements4 reported findingsCorrect

The dual-graph construction of natural extensions, the invariant-density formulas for the listed multidimensional continued-fraction algorithms, and the ergodicity conclusions are supported.

Proposition 3.2 and Lemmas 3.4–3.5Correct

Dual labeling produces a measure-preserving natural extension

Pages 7–11 · Section 3 · arXiv:2606.03766v1

Equation (2), $M_e^T\Omega_v=\Omega_{v'}M_{e'}$ , is exactly the condition that the vertex-dependent bilinear form be preserved. When the $\Omega_v$ are nonsingular, the change of variables $a\mapsto\Omega_va$ conjugates the graph extension to the Arnoux–Labbé extension on the union of the cones $\Omega_v\mathbb R_+^{d+1}$ . The determinant-one branch maps preserve ambient Lebesgue measure, and radial disintegration on $\langle x,a\rangle_v=1$ gives the invariant hypersurface measure and its projective marginal.

Propositions 5.3 and 5.6Correct

Invariant densities for Selmer and Triangle algorithms

Pages 17–26 · Section 5 · arXiv:2606.03766v1

The explicit dual labels solve Equation (2), and summing the cone integrals over the graph fibers gives the stated homogeneous densities. The projection matrices are unimodular on each branch, so no additional Jacobian factor is missing. First-return acceleration sums exactly the branches whose denominators combine into $(x_{\sigma0}+x_{\sigma d})x_{\sigma0}\cdots x_{\sigma,d-1}$ .

Corollaries 7.5, 7.7, 7.8 and 7.11Correct

Ergodicity and uniqueness in the Lebesgue class

Pages 35–37 · Section 7 · arXiv:2606.03766v1

The graph checks verify both clauses of Fougeron's nondegeneration criterion for the Triangle and Modified Triangle systems; the Selmer case is imported from the cited proposition. The criterion yields uniqueness and ergodicity within the Lebesgue measure class. Passing to the first-return acceleration preserves that conclusion on the accelerated domain.

Section 6.3Correct

Brun invariant densities in dimensions two and three

Pages 28–33 · Section 6.3 · arXiv:2606.03766v1

The first return of the permutation graph is conjugate to the Brun algorithm. In dimension two, the six vertex-cone contributions sum to $1/(6x_{\sigma0}x_{\sigma1}(x_{\sigma0}+x_{\sigma2}))$ on each ordering cone. In dimension three the one-copy equation has no solution, but the displayed two-extension supplies two valid dual matrices per permutation; summing all twelve cone contributions gives the stated homogeneous density. The manuscript presents the later Modified Triangle fractal description only as a conjecture, not as a proved density formula.

02Proofs2 reported findingsCorrect

The graph conjugacy, cone integration, explicit matrix calculations, and nondegeneration arguments are correct and complete.

Sections 3, 5 and 7Correct and complete

Natural-extension, density, and ergodicity proof chains

Pages 7–37 · arXiv:2606.03766v1

Each branch of the dual graph is paired with exactly one inverse branch of the original graph, which proves bijectivity rather than only invariance. The density calculations integrate over all and only the cones in the corresponding vertex fiber. In Section 7, every strongly connected component of the restricted graphs is covered by either the one-outgoing-label alternative or an explicit path that exits the component, so the cited ergodicity theorem applies.

Definition 6.2 and Sections 6.2–6.4Correct and complete

First-return dual graphs and finite extensions

Pages 27–34 · arXiv:2606.03766v1

Replacing edges by return paths preserves the matrix-product identities used in Equation (2), so a $W$ -dual graph supplies inverse branches for the first-return induction itself. The Brun constructions explicitly check these identities; the two-sheet extension in dimension three records which of the two dual cones contains the fiber point and hence avoids double counting. The Modified Triangle discussion correctly stops where no finite extension or invariant domain has been proved.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2606.03766v1
Authors listed: David Siukaev
Audit date: August 18, 2026

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A Graph-Based Method for Invariant Densities of Multidimensional Continued Fractions

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Dual labeling produces a measure-preserving natural extension

Invariant densities for Selmer and Triangle algorithms

Ergodicity and uniqueness in the Lebesgue class

Brun invariant densities in dimensions two and three

Natural-extension, density, and ergodicity proof chains

First-return dual graphs and finite extensions

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