On Stronger Forms of Expansivity

We define the concept of stronger forms of positively expansive map and name it as $p \:\mathscr{F}-$expansive maps. Here $\mathscr{F}$ is a family of subsets of $\mathbb{N}$. Examples of positively thick expansive and positively syndetic expansive maps are constructed here. Also, we obtain conditions under which a positively expansive map is positively co--finite expansive and positively syndetic expansive maps. Further, we study several properties of $p \:\mathscr{F}-$expansive maps. A characterization of $p \:\mathscr{F}-$expansive maps in terms of $p \:\mathscr{F}^*-$generator is obtained. Here $p \:\mathscr{F}^*$ is dual of $\mathscr{F}$. Considering $(\mathbb{Z},+)$ as a semigroup, we study $\mathscr{F}-$expansive homeomorphism, where $\mathscr{F}$ is a family of subsets of $\mathbb{Z} \setminus \{0\}$. We show that there does not exists an expansive homeomorphism on a compact metric space which is $\mathscr{F}_s-$expansive. Also, we study relation between $\mathscr{F}-$expansivity of $f$ and the shift map $\sigma_f$ on the inverse limit space.