On Stronger Forms of Expansivity

We define the concept of stronger forms of positively expansive map and name it as p \:F-expansive maps. Here F is a family of subsets of 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 \:F-expansive maps. A characterization of p \:F-expansive maps in terms of p \:F^*-generator is obtained. Here p \:F^* is dual of F. Considering (Z,+) as a semigroup, we study F-expansive homeomorphism, where F is a family of subsets of Z \0\. We show that there does not exists an expansive homeomorphism on a compact metric space which is Fₛ-expansive. Also, we study relation between F-expansivity of f and the shift map σf on the inverse limit space.