Asymptotic fixed point theory and the beer barrel theorem

In Sections 2 and 3 of this paper we refine and generalize theorems of Nussbaum (see [42]) concerning the approximate fixed point index and the fixed point index class. In Section 4 we indicate how these results imply a wide variety of asymptotic fixed point theorems. In Section 5 we prove a generalization of the mod p theorem: if p is a prime number, f belongs to the fixed point index class and f satisfies certain natural hypothesis, then the fixed point index of f ^p is congruent mod p to the fixed point index of f. In Section 6 we give a counterexample to part of an asymptotic fixed point theorem of A. Tromba [55]. Sections 2, 3, and 4 comprise both new and expository material. Sections 5 and 6 comprise new results.