Positively weak measure expansive differentiable maps
Bull. Korean Math. Soc. 2020 Vol. 57, No. 3, 569-581
https://doi.org/10.4134/BKMS.b181190
Published online May 31, 2020
Jiweon Ahn, Manseob Lee
Chungnam National University; Mokwon University
Abstract : In this paper, we introduce the new general concept of usual expansiveness which is called ``positively weak measure expansiveness" and study the basic properties of positively weak measure expansive $C^1$-differentiable maps on a compact smooth manifold $M$. And we prove that the following theorems. \begin{itemize} \item[(1)] Let $\mathcal{PWE}$ be the set of all positively weak measure expansive differentiable maps of $M$. Denote by $\rm{int}(\mathcal{PWE})$ is a $C^1$-interior of $\mathcal{PWE}$. $f \in\rm{int}(\mathcal{PWE})$ if and only if $f$ is expanding. \item[(2)] For $C^1$-{\it generic} $f \in C^1(M)$, $f$ is positively weak measure-expansive if and only if $f$ is expanding. \end{itemize}
Keywords : Expansive, measure expansive, weak measure expansive, positively weak measure expansive, expanding
MSC numbers : Primary 37C20, 37D20
Supported by : The first author was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government (Ministry of Education)(No.NRF-2017R1D1A1B03032106).
The second author was supported by the National Research Foundation of Korea(NRF) by the Korea government (MSIP) (No. NRF-2017R1A2B4001892).
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