Volume preserving dynamics without genericity and related topics
Commun. Korean Math. Soc. 2012 Vol. 27, No. 2, 369-375
https://doi.org/10.4134/CKMS.2012.27.2.369
Printed June 1, 2012
Jaeyoo Choy, Hahng-Yun Chu, and Min Kyu Kim
Kyungpook National University, Chungnam National University, Gyeongin National University of Education
Abstract : In this article, we focus on certain dynamic phenomena in volume-preserving manifolds. Let $M$ be a compact manifold with a volume form $\omega$ and $f : M \rightarrow M$ be a diffeomorphism of class $\mathcal{C}^1$ that preserves $\omega$. In this paper, we do $not$ assume $f$ is $\mathcal{C}^1$-generic. We prove that $f$ satisfies the chain transitivity and we also show that, on $M$, the $\mathcal{C}^1$-stable shadowability is equivalent to the hyperbolicity.
Keywords : hyperbolicity, $\mathcal{C}^1$-stable shadowable, chain recurrence, chain transitive
MSC numbers : Primary 37C50; Secondary 37C70, 54H20
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