On Stronger Forms of Expansivity

We define the concept of stronger forms of positively expansive map and name it as $p\mathcal{F}-$expansive maps. Here $\mathcal{F}$ 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 $p\mathcal{F}-$expansive maps. A characterization of $p\mathcal{F}-$expansive maps in terms of $p{\mathcal{F}}^{\ast }-$generator is obtained. Here $p{\mathcal{F}}^{\ast }$ is dual of $\mathcal{F}$. Considering $(\mathbb{Z},+)$ as a semigroup, we study $\mathcal{F}-$expansive homeomorphism, where $\mathcal{F}$ 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 ${\mathcal{F}}_s-$expansive. Also, we study relation between $\mathcal{F}-$expansivity of $f$ and the shift map ${\sigma }_f$ on the inverse limit space.