Periodic points on T-fiber bundles over the circle (Q1704389)

This paper is concerned with periodic points of iterated fiber-preserving maps on fiber bundles with base \(S^1\) and fiber the torus (i.e. the quotient space \(\mathbb R\times\mathbb R/\mathbb Z\times\mathbb Z\)). Given a fiber preserving map \(f\) and a positive integer \(n\), necessary and sufficient conditions are given such that \(f^n\) can be deformed to a fixed-point-free map on \(S^1\). This is applied to find a map \(g\) fiberwise homotopic to \(f\) such that \(g^n\) is a fixed-point-free map over \(S^1\).