Published paper
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:
- Proposition 5.1 and Corollary 5.3 · Annals pp. 529–532The Section 5 construction proves that the selected increasing partition subordinate to the full unstable foliation captures the full metric entropy.
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Exact reviewed source
Version of record · Annals of Mathematics 122 (1985), no. 3, 509–539
François Ledrappier and Lai-Sang Young. The metric entropy of diffeomorphisms. Part I: Characterization of measures satisfying Pesin's entropy formula. Annals of Mathematics 122 (1985), no. 3, 509–539.
Open audited source ↗01Statements1 reported findingNo statement issues found
The selected Section 5 entropy proposition and its subordinate-partition corollary used by Part II are correct.
Proposition 5.1 and Corollary 5.3No issue found
A partition subordinate to the full unstable foliation captures the full metric entropy
VOR pp. 529–532 · Proposition 5.1 and Corollary 5.3
Proposition 5.1 constructs the increasing measurable partition used to approximate the entropy from below through unstable transverse dimensions. Letting the error parameter tend to zero gives Corollary 5.3: every partition of the specified Section 3.1 type subordinate to the full unstable foliation satisfies . This is the precise Part I result invoked for the terminal partial-entropy identity in Part II; the broader Part I Theorems A–B are not selected dependencies.
02Proofs2 reported findingsNo proof issues found
The five-step Section 5 argument closes through the represented Lyapunov-chart, adapted-partition, subordinate-disk, stable-manifold, and covering inputs.
Proof of Proposition 5.1No issue found in the reviewed proof
Conditional ball estimates yield the required entropy lower bound
VOR pp. 529–532 · proof parts (A)–(E)
The proof fixes Lyapunov charts and a finite-entropy adapted partition, constructs nested unstable partitions, and compares their conditional measures on transverse balls. Lemma 5.2 propagates the balls between returns to a regular block; the integrable maximal function from Section 4 controls the exceptional contribution. Birkhoff averages then give the printed lower bound, and Corollary 5.3 follows by letting the error parameter tend to zero. The indices, return-time estimate, and entropy limits agree.
Recursive source expansionCorrect source use
Only the proof-critical foundations for Proposition 5.1 are represented
VOR Sections 2–5 and Appendix; cited Oseledets 1968, Mañé 1981, Ledrappier–Strelcyn 1982, Fathi–Herman–Yoccoz LNM 1007, and de Guzmán LNM 481
Oseledets supplies the regular splitting and tempered coordinates, Mañé supplies the finite-entropy adapted partition, and Ledrappier–Strelcyn supplies the measurable subordinate disk family. The stable-manifold input is terminalized at the Fathi–Herman–Yoccoz monograph chapter, and the bounded-overlap selection is terminalized at de Guzmán's monograph. The Ruelle-inequality and Pesin-formula branches used only for Part I Theorems A–B are outside this selected claim and are not drawn as dependencies.
03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.