Asymptotic fixed point theory and the beer barrel theorem (Q841654)

In the present paper, concerning the approximate fixed point index and the fixed point index class, a number of refinements and generalizations of theorems in [\textit{R.\,D.\thinspace Nussbaum}, ``Generalizing the fixed point index'', Math.\ Ann.\ 228, 259--278 (1977; Zbl 0365.58005)] are introduced, together with a broad range of examples. It is shown how these results imply a wide variety of asymptotic fixed point theorems. Next, the authors 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 a certain natural hypothesis, then the fixed point index of \(f^p\) is congruent mod \(p\) to the fixed point index of \(f\). Moreover, a counterexample to part of an asymptotic fixed point theorem of \textit{A.\,J.\thinspace Tromba} [``A general asymptotic fixed point theorem'', J.~Reine Angew.\ Math.\ 332, 118--1125 (1982; Zbl 0501.58009)] is given.