Generic Kleene fixed point theorem

Kleene's fixed point theorem states that any algorithmic mapping $\mathcal{A}$ of the set of Turing machines to the set of Turing machines has a fixed point: there is a Turing machine $M$ such that machine $\mathcal{A}(M)$ computes the same function as $M$. In this paper we prove a generic analog of this theorem: any algorithmic mapping $\mathcal{A}$ of a set of “almost all” Turing machines to the set of Turing machines has a fixed point.