Positively weak measure expansive differentiable maps
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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}
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}
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