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A Pasting Lemma I: the case of vector fields

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Alexander Arbieto, Carlos Matheus

Dynamical Systems · Submitted 14 Oct 2003

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We prove a perturbation (pasting) lemma for conservative (and symplectic) systems.

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arXiv IDmath/0310207 (version 1)
Pith ID6WPCYS52, Pith's permanent record of this paper and its review
CitationAlexander Arbieto, Carlos Matheus. "A Pasting Lemma I: the case of vector fields." arXiv:math/0310207 (2003). https://pith.science/paper/6WPCYS52
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,476 input tokens.

What the abstract itself states, as probabilities: claims first 0.10 · states limitation 0.71 · human participants 0.02 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.94 · mentions code or data availability 0.01.

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arXiv listing abstract v1. Results and the main claim are underlined; .

We prove a perturbation (pasting) lemma for conservative (and symplectic) systems. This allows us to prove that $C^{\infty}$ volume preserving vector fields are $C^1$-dense in $C^{1}$ volume preserving vector fields (After the conclusion of this work, Ali Tahzibi pointed out to us that this result was proved in 1979 by Carlos Zuppa, although his proof is different from ours.). Moreover, we obtain that $C^1$ robustly transitive conservative flows in three-dimensional manifolds are Anosov and we conclude that there are no geometrical Lorenz-like sets for conservative flows. Also, by-product of the version of our pasting lemma for conservative diffeomorphisms, we show that $C^1$-robustly transitive conservative $C^2$-diffeomorphisms admits a dominated splitting, thus solving a question posed by Bonatti-Diaz-Pujals. In particular, stably ergodic diffeomorphisms admits a dominated splitting.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We prove a perturbation (pasting) lemma for conservative (and symplectic) systems. contribution 0.94 0.96 0.71 asserted 1.00
s1 This allows us to prove that $C^{\infty}$ volume preserving vector fields are $C^1$-dense in $C^{1}$ volume preserving vector fields (After the conclusion of this work, Ali Tahzibi pointed out to us that this result was proved in 1979 by Carlos Zuppa, although his proof is different from ours.). result 0.91 0.76 0.24 asserted 1.00
s2 Moreover, we obtain that $C^1$ robustly transitive conservative flows in three-dimensional manifolds are Anosov and we conclude that there are no geometrical Lorenz-like sets for conservative flows. result 0.99 0.94 0.24 asserted 1.00
s3 Also, by-product of the version of our pasting lemma for conservative diffeomorphisms, we show that $C^1$-robustly transitive conservative $C^2$-diffeomorphisms admits a dominated splitting, thus solving a question posed by Bonatti-Diaz-Pujals. result 0.96 0.95 0.20 asserted 1.00
s4 In particular, stably ergodic diffeomorphisms admits a dominated splitting. result 0.90 0.87 0.12 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We prove a perturbation (pasting) lemma for conservative (and symplectic) systems. This allows us to prove that $C^{\infty}$ volume preserving vector fields are $C^1$-dense in $C^{1}$ volume preserving vector fields (After the conclusion of this work, Ali Tahzibi pointed out to us that this result was proved in 1979 by Carlos Zuppa, although his proof is different from ours.). Moreover, we obtain that $C^1$ robustly transitive conservative flows in three-dimensional manifolds are Anosov and we conclude that there are no geometrical Lorenz-like sets for conservative flows. Also, by-product of the version of our pasting lemma for conservative diffeomorphisms, we show that $C^1$-robustly transitive conservative $C^2$-diffeomorphisms admits a dominated splitting, thus solving a question posed by Bonatti-Diaz-Pujals. In particular, stably ergodic diffeomorphisms admits a dominated splitting.

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