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pith:JKN3IHOS

pith:2026:JKN3IHOSQXYXQL2C7QFZOTYA22

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Continuity properties of partial entropy

Gang Liao, Huirong Tao, Jiagang Yang, Yao Tong

For C^{1+α} diffeomorphisms, partial entropy in every direction is upper semi-continuous whenever the sums of Lyapunov exponents are continuous.

arxiv:2605.13273 v1 · 2026-05-13 · math.DS

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Claims

C1strongest claim

We establish a general criterion on the upper semi-continuity of partial entropy in all directions for C^{1+α} diffeomorphisms: it holds when the respective sums of Lyapunov exponents are continuous.

C2weakest assumption

The diffeomorphism must be C^{1+α} and the sums of Lyapunov exponents must be continuous; the criterion fails to apply if either condition is dropped.

C3one line summary

Partial entropies are upper semi-continuous for C^{1+α} diffeomorphisms when Lyapunov exponent sums are continuous, implying the same property at generic ergodic measures.

References

91 extracted · 91 resolved · 2 Pith anchors

[1] F. Abdenur, C. Bonatti, and S. Crovisier. Nonuniform hyperbolicity forC 1-generic diffeomor- phisms.Israel J. Math., 183:1–60, 2011 2011

[2] V. S. Afraimovich, V. V. Bykov, and L. P. Shilnikov. The origin and structure of the Lorenz attractor.Dokl. Akad. Nauk SSSR, 234(2):336–339, 1977 1977

[3] V. Araujo, M. J. Pacifico, E. R. Pujals, and M. Viana. Singular-hyperbolic attractors are chaotic.Trans. Amer. Math. Soc., 361(5):2431–2485, 2009 2009

[4] A. Avila and M. Viana. Extremal Lyapunov exponents: an invariance principle and applications. Invent. Math., 181(1):115–189, 2010 2010

[5] On the loss of upper semi-continuity of metric entropy for $C^{r}$ diffeomorphisms 2026 · arXiv:2604.05611

Receipt and verification

First computed	2026-05-18T02:44:49.259585Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

4a9bb41dd285f1782f42fc0b974f00d697c023c353ba1586b85d315493cd8577

Aliases

arxiv: 2605.13273 · arxiv_version: 2605.13273v1 · doi: 10.48550/arxiv.2605.13273 · pith_short_12: JKN3IHOSQXYX · pith_short_16: JKN3IHOSQXYXQL2C · pith_short_8: JKN3IHOS

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curl -sH 'Accept: application/ld+json' https://pith.science/pith/JKN3IHOSQXYXQL2C7QFZOTYA22 \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 4a9bb41dd285f1782f42fc0b974f00d697c023c353ba1586b85d315493cd8577

Canonical record JSON

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  "metadata": {
    "abstract_canon_sha256": "14baa8965897cf8d38c8b8e01faaf35e3c4e5c2da761d4a391d46b71815d33ba",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.DS",
    "submitted_at": "2026-05-13T09:50:11Z",
    "title_canon_sha256": "9d7d626163fbd111f2c9ded67ff04d3b5ade75b3c13253d18c7f783766417585"
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    "kind": "arxiv",
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