Properties of generic and almost every mappings in ZZ

In a Polish group G, a property is said to hold for a generic g∈G if the set on which it does not hold is meager in G; a property holds for almost every (ae) g∈G if the set on which it does not hold is Haar null. In this paper we study the properties of generic and ae elements of the Baer–Specker Group ZZ, the space of all self-maps of Z. We find that with respect to most of the properties under consideration in this paper, the properties of a generic element and ae element are complementary. Our results are similar in flavor to results by Dougherty and Mycielski for the group S∞ of all permutations of N.