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no. 2
Sufficient conditions for rescaling expansivity
Arnoldo Rojas1; Xiao Wen2; Yinong Yang3
1 Beijing Advanced Innovation Center for Future Blockchain and Privacy Computing, Beihang University, Beijing, China & Beijing Academy of Blockchain and edge Computing, Beijing, 100086, China
2 LMIB, Institute of Artificial Intelligence and School of Mathematical Science, Beihang University, Beijing, 100086, China
3 School of Mathematics and Statistics, Liaoning University, Shenyang 110036, China
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 33 (2024) no. 2, pp. 447-467.

We show that every k*-expansive vector field of a closed manifold is rescaling expansive. This improves the main result in [5]. The proof relies on the new notion of singular-expansive flow which will be studied.

Nous montrons que tout champ de vecteurs k*-expansif d’une variété fermée est rééchelonné expansif. Cela améliore le résultat principal dans [5]. La preuve repose sur la nouvelle notion de flux singulier-expansif qui sera étudiée.

Received:
Accepted:
Published online:
DOI: 10.5802/afst.1777
Classification: 37C10, 37B05
Keywords: Singular-expansive, k*-expansive, Rescaling expansive

Arnoldo Rojas 1; Xiao Wen 2; Yinong Yang 3

1 Beijing Advanced Innovation Center for Future Blockchain and Privacy Computing, Beihang University, Beijing, China & Beijing Academy of Blockchain and edge Computing, Beijing, 100086, China
2 LMIB, Institute of Artificial Intelligence and School of Mathematical Science, Beihang University, Beijing, 100086, China
3 School of Mathematics and Statistics, Liaoning University, Shenyang 110036, China
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Arnoldo Rojas; Xiao Wen; Yinong Yang. Sufficient conditions for rescaling expansivity. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 33 (2024) no. 2, pp. 447-467. doi : 10.5802/afst.1777. https://afst.centre-mersenne.org/articles/10.5802/afst.1777/

[1] Valentin S. Afrajmovich; V. V. Bykov; Leonid P. Shil’nikov The origin and structure of the Lorenz attractor, Dokl. Akad. Nauk SSSR, Volume 234 (1977) no. 2, pp. 336-339 | Zbl

[2] Vitor Araujo; Maria J. Pacifico; Enrique R. Pujals; Marcelo Viana Singular-hyperbolic attractors are chaotic, Trans. Am. Math. Soc., Volume 361 (2009) no. 5, pp. 2431-2485 | DOI | Zbl

[3] Alfonso Artigue Expansive flows of surfaces, Discrete Contin. Dyn. Syst., Volume 33 (2013) no. 2, pp. 505-525 | DOI | Zbl

[4] Alfonso Artigue Expansive flows of the three-sphere, Differ. Geom. Appl., Volume 41 (2015), pp. 91-101 | DOI | Zbl

[5] Alfonso Artigue Rescaled expansivity and separating flows, Discrete Contin. Dyn. Syst., Volume 38 (2018) no. 9, pp. 4433-4447 | DOI | Zbl

[6] Joseph Auslander Minimal flows and their extensions, North-Holland Mathematics Studies, 153, North-Holland, 1988

[7] Christian Bonatti; Adriana da Luz Star flows and multisingular hyperbolicity (2018) | arXiv

[8] Wescley Bonomo; Jorge Rocha; Paulo Varandas The centralizer of Komuro-expansive flows and expansive ℝ d actions, Math. Z., Volume 289 (2018) no. 3-4, pp. 1059-1088 | DOI | Zbl

[9] Rufus Bowen Entropy for group endomorphisms and homogeneous spaces, Trans. Am. Math. Soc., Volume 153 (1971), pp. 401-414 | DOI | Zbl

[10] Rufus Bowen; Peter Walters Expansive one-parameter flows, J. Differ. Equations, Volume 12 (1972), pp. 180-193 | DOI | Zbl

[11] John Guckenheimer A strange, strange attractor, The Hopf bifurcation theorem and its applications, Springer, 1976, pp. 368-381 | DOI

[12] Motomasa Komuro Expansive properties of Lorenz attractors, The theory of dynamical systems and its applications to nonlinear problems (Kyoto, 1984), World Scientific, 1984, pp. 4-26

[13] Motomasa Komuro One-parameter flows with the pseudo-orbit tracing property, Monatsh. Math., Volume 98 (1984) no. 3, pp. 219-253 | DOI | Zbl

[14] Masatoshi Oka Expansivity of real flows, Tsukuba J. Math., Volume 14 (1990) no. 1, pp. 1-8 | Zbl

[15] Jacob Palis; Welington de Melo Geometric theory of dynamical systems. An introduction, Springer, 1982 | DOI

[16] Winfried R. Utz Unstable homeomorphisms, Proc. Am. Math. Soc., Volume 1 (1950), pp. 769-774 | Zbl

[17] Xiao Wen; Lan Wen A rescaled expansiveness for flows, Trans. Am. Math. Soc., Volume 371 (2019) no. 5, pp. 3179-3207 | Zbl

[18] Xiao Wen; Yining Yu Equivalent definitions of rescaled expansiveness, J. Korean Math. Soc., Volume 55 (2018) no. 3, pp. 593-604 | Zbl

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