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Publication metadata

Journal citation
Communications on Pure and Applied Mathematics 56(8), 1184–1221 (2003)
Publication date
2003

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 statements:

  • Lemma 6.1 · Author manuscript PDF pp. 16–18Noncommuting root conditionals propagate translation invariance.
  • Lemma 6.2 · Author manuscript PDF pp. 18–19Entropy splits into the weighted sum of root contributions.
  • Corollary 7.2 · Author manuscript PDF pp. 22–24Ledrappier–Young's formula applies to the one-dimensional root factors.
  • Proposition 8.3 · Author manuscript PDF pp. 27–31Conditional measures on an unstable subgroup factor over the ordered root foliations.

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Author-hosted manuscript · explicit fallback for inaccessible version of record

Manfred Einsiedler and Anatole Katok. Invariant measures on G/ΓG/\Gamma for split simple Lie groups GG. Communications on Pure and Applied Mathematics 56 (2003), no. 8, 1184–1221.

The proof-bearing author manuscript with the final article structure was openly accessible; no Wiley VOR download is claimed.

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Generated August 27, 2026
01Statements2 reported findingsNo statement issues found

The selected conditional-measure commutator and entropy-product statements are correct.

Lemmas 6.1–6.2No issue found

Nonatomic root conditionals propagate and entropy splits by roots

Author manuscript PDF pp. 16–19 · Lemmas 6.1–6.2

Lemma 6.1 uses noncommuting root subgroups to propagate translation invariance of conditional measures. Lemma 6.2 expresses entropy as the weighted sum of root contributions. These are exactly the facts used inside EKL's high-entropy measure classification.

Proposition 8.3No issue found

Unstable conditional measures factor over the selected root foliations

Author manuscript PDF pp. 27–31 · Proposition 8.3

For a closed root set, the conditional measure along its unstable subgroup is equivalent to the ordered product of the one-dimensional root conditionals, matching EKL's use.

02Proofs2 reported findingsNo proof issues found

The commutator and product-measure arguments are complete; the entropy identity is correctly reduced to Ledrappier–Young.

Lemma 6.1No issue found in the reviewed proof

The commutator calculation produces a dense translation subgroup

Author manuscript PDF pp. 16–18

The proof tracks conditional measures under two root translations, compares the two commutator orders, and obtains invariance under a nontrivial element of the target root group. Recurrence and scaling make the generated subgroup dense, forcing Haar conditionals.

Lemma 6.2 and Proposition 8.3No issue found in the reviewed proof

The entropy formula and leafwise product decomposition align

Author manuscript PDF pp. 18–19 and 27–35 · Corollary 7.2 and Proposition 8.3

Proposition 8.3 is proved by successive conditioning on a normal root ordering. Corollary 7.2 then applies Ledrappier–Young Theorem C to those one-dimensional factors; its displayed coefficients are the Lyapunov exponents, yielding Lemma 6.2.

03Novelty0 reported findingsNo novelty concerns found

No non-novelty findings.

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