On stronger forms of expansive maps

Abstract

We define the concept of stronger forms of positively expansive maps and name them $p\mathscr{F}$-expansive maps. Here $\mathscr{F}$ is a family of subsets of $\mathbb{N}$. Examples of $p\mathscr{F}$-expansive maps, for $\mathscr{F}=\mathscr{F}_t, \mathscr{F}_s, \mathscr{F}_{cf}$, are discussed. Also, we give examples of maps which are $p\mathscr{F}_t$-expansive but not $p\mathscr{F}_{cf}$-expansive, $p\mathscr{F}_s$-expansive but not $p\mathscr{F}_t$-expansive, and $p\mathscr{F}_s$-expansive but not $p\mathscr{F}_{cf}$-expansive. Further, we obtain conditions under which a positively expansive map is $p\mathscr{F}_{cf}$-expansive and $p\mathscr{F}_s$-expansive map. We next study several properties of $p\mathscr{F}$-expansive maps. A characterization of $p\mathscr{F}$-expansive maps in terms of $\mathscr{F}^*$-generator is obtained. Here $\mathscr{F}^*$ is dual of $\mathscr{F}$. Considering $(\Z,+)$ as a semigroup, we study $\mathscr{F}$-expansive homeomorphism, where $\mathscr{F}$ is a family of subsets of $\Z \setminus \{0\}$. We show that there does not exist 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.