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Role in dependence graphs
Proof-critical source
Winning property of counterexamples to Uniform Littlewood's Conjecture
This paper is included only for the following marked statement:
- Theorems 1–4 · Russian VOR pp. 193–194; proof §4 pp. 197–210Supplies the regular Lyapunov splitting and tempered coordinates used in Ledrappier–Young's selected Section 5 argument.
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Exact MathNet scan of the Russian version of record · Trudy Moskovskogo Matematicheskogo Obshchestva 19 (1968), 179–210
V. I. Oseledets. A multiplicative ergodic theorem. Characteristic Ljapunov exponents of dynamical systems. Trudy Moskovskogo Matematicheskogo Obshchestva 19 (1968), 179–210 (Russian); English translation, Transactions of the Moscow Mathematical Society 19 (1968), 197–231.
Open audited source ↗01Statements1 reported findingNo statement issues found
The multiplicative ergodic theorem supplies the almost-everywhere invariant Lyapunov splitting and exponential rates used in Ledrappier–Young's regular-point and chart construction.
Theorems 1–4No issue found
Integrable linear cocycles have almost-everywhere Lyapunov data
Russian VOR pp. 193–194 · Theorems 1–4; proof § 4, pp. 197–210
Under the printed logarithmic integrability condition, Theorems 1–4 give Lyapunov regularity, limits of all exterior-power growth rates and singular values, and a measurable invariant direct-sum decomposition with uniform exponential rates on each Lyapunov subspace. Applied to the derivative cocycle of a smooth diffeomorphism, these are the regular splitting, exponents, and tempered-coordinate inputs used in the Ledrappier–Young Appendix and Sections 2–5.
02Proofs2 reported findingsNo proof issues found
The cocycle reductions and fixed-point construction close through the represented pointwise theorem and its exact modern monograph formulation.
Section 4 proofCorrect and complete modulo represented source
The triangular-cocycle and invariant-measure argument is coherent
Russian VOR pp. 197–210 · § 4
The proof first treats determinant growth by additive cocycles, reduces continuous time to discrete time, proves the triangular case, and then uses compactness and a fixed-point argument to obtain an invariant measure on the orthogonal-group extension. A generic fiber point satisfying the scalar ergodic averages completes the reduction. The exterior-power and invariant-splitting conclusions are consistent with the stated integrability hypotheses.
Recursive source expansionClosed with represented article branch
Birkhoff's pointwise ergodic theorem is the only indispensable ordinary-paper input
Russian VOR p. 198 and p. 209 · explicit Birkhoff applications and final proof summary
Oseledets applies the pointwise ergodic theorem to logarithms of diagonal cocycle entries and says in the final paragraph that this is the only essential external fact. Birkhoff's 1931 paper is represented with its exact historical scope, and Peter Walters, An Introduction to Ergodic Theory, Theorem 1.14, pp. 34–35, supplies the terminal monograph statement and proof in the arbitrary probability-preserving form used here. Lyapunov's basis construction, Tychonoff's fixed-point theorem, and weak compactness are terminal standard foundations cited through the printed books; Furstenberg–Kesten is discussed as a special case, not imported into the proof.
03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.