Published paper

A proof of Pesin's formula

Ricardo Mañé

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Journal citation: Ergodic Theory and Dynamical Systems 1(1), 95–102 (1981)
Publication date: 1981
DOI: 10.1017/S0143385700001188 ↗

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:

Lemma 2 · VOR pp. 96–97; 1983 erratum checkedConstructs a finite-entropy partition with atom diameters adapted to the Lyapunov regularity scale.

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Cambridge version of record · Ergodic Theory and Dynamical Systems 1 (1981), 95–102

Ricardo Mañé. A proof of Pesin's formula. Ergodic Theory and Dynamical Systems 1 (1981), 95–102.

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Generated August 28, 2026

01Statements1 reported findingNo statement issues found

Lemma 2 gives exactly the finite-entropy, pointwise-diameter partition used in Ledrappier–Young's adapted-partition construction.

Lemma 2No issue found

An integrable logarithmic radius admits a countable finite-entropy refinement

Version of record pp. 96–97 · Lemma 2 and proof

If $\mu$ is a probability measure and $\rho:M\to(0,1)$ has integrable $\log\rho$ , Lemma 2 constructs a countable partition of finite entropy whose atom at almost every $x$ has diameter smaller than $\rho(x)$ . Ledrappier–Young invokes this lemma in Lemma 2.4.2 to make its regularity-adapted partition finite-entropy.

02Proofs2 reported findingsNo proof issues found

The selected lemma is proved directly by dyadic-level decomposition and elementary entropy estimates.

Lemma 2 proofNo issue found in the reviewed proof

The covering and summability estimates close internally

Version of record pp. 96–97 · Lemmas 1–2

The proof decomposes points by exponential levels of $\rho$ , refines each level by a bounded-cardinality small-diameter partition, and bounds the resulting entropy using the integrability sum and the elementary Lemma 1. No later use of Shannon–McMillan–Breiman, stable manifolds, or the Pesin-formula argument is needed for this selected claim, and no further ordinary-paper parent is opened.

Correction checkCorrect and unaffected by known corrections

The published 1983 erratum does not affect Lemma 2

Erratum DOI 10.1017/S0143385700001863 checked against selected locator

The later correction concerns a different part of the article and does not modify the statement or proof of Lemma 2 on pages 96–97.

03Novelty0 reported findingsNo novelty concerns found

No non-novelty findings.

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