TY - JOUR

T1 - On two notions of expansiveness for continuous semiflows

AU - Herrero, Sebastián

AU - Jaque, Nelda

N1 - Funding Information:
The authors would like to thank C.A. Morales for sharing with them a copy of the preprint [6] . The authors also thank the anonymous referee for her/his helpful comments that improved the quality of the manuscript. S. Herrero was partially supported by Proyecto VRIEA-PUCV 039.344/2021 . N. Jaque was partially supported by ANID/CONICYT FONDECYT Iniciación grant 11190815 .
Publisher Copyright:
© 2022 Elsevier Inc.

PY - 2022/11/1

Y1 - 2022/11/1

N2 - We study two notions of expansiveness for continuous semiflows: expansiveness in the sense of Alves, Carvalho and Siqueira (2017), and an adaptation of positive expansiveness in the sense of Artigue (2014). We prove that if X is a metric space and ϕ is an expansive semiflow on X according to the first definition, then the semiflow ϕ is trivial and the space X is uniformly discrete. In particular, if X is compact then it is finite. With respect to the second definition, we prove that if X is a compact metric space and ϕ is a positive expansive semiflow on it, then X is a union of at most finitely many closed orbits, unbranched tails and isolated singularities.

AB - We study two notions of expansiveness for continuous semiflows: expansiveness in the sense of Alves, Carvalho and Siqueira (2017), and an adaptation of positive expansiveness in the sense of Artigue (2014). We prove that if X is a metric space and ϕ is an expansive semiflow on X according to the first definition, then the semiflow ϕ is trivial and the space X is uniformly discrete. In particular, if X is compact then it is finite. With respect to the second definition, we prove that if X is a compact metric space and ϕ is a positive expansive semiflow on it, then X is a union of at most finitely many closed orbits, unbranched tails and isolated singularities.

KW - Continuous semiflows

KW - Expansiveness

UR - http://www.scopus.com/inward/record.url?scp=85131673416&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2022.126405

DO - 10.1016/j.jmaa.2022.126405

M3 - Article

AN - SCOPUS:85131673416

VL - 515

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 1

M1 - 126405

ER -