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J. Korean Math. Soc. 2023; 60(5): 987-998
Online first article August 17, 2023 Printed September 1, 2023
https://doi.org/10.4134/JKMS.j220359
Copyright © The Korean Mathematical Society.
Continuum-wise expansiveness for $C^1$ generic vector fields
Manseob Lee
Mokwon University
It is shown that every continuum-wise expansive $C^1$ generic vector field $X$ on a compact connected smooth manifold $M$ satisfies Axiom A and has no cycles, and every continuum-wise expansive homoclinic class of a $C^1$ generic vector field $X$ on a compact connected smooth manifold $M$ is hyperbolic. Moreover, every continuum-wise expansive $C^1$ generic divergence-free vector field $X$ on a compact connected smooth manifold $M$ is Anosov.
Keywords: Expansive, continuum-wise expansive, Axiom A, homoclinic class, hyperbolic, generic
MSC numbers: 37C10, 37C20, 34D10, 37C27
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