Homotopy minimal periods for maps of three-dimensional solvmanifolds
A natural number \(m\) is said to be a homotopy minimal period of a map \(f\colon X\to X\) if each map in the homotopy class of \(f\) has periodic points of minimal period \(m\). The authors of this paper give a description of the sets of homotopy minimal periods of maps of all compact \(3\)-dimensional solvmanifolds. It was noted by \textit{G. D. Mostow} [Ann. Math. (2) 60, 1--27 (1954; Zbl 0057.26103)] that every solvmanifold \(M\) has a fibration structure, called minimal Mostow fibration, such that every map on \(M\) is homotopic to a fibre-preserving map of this fibration. A \(3\)-dimensional solvmanifold is either a circle bundle over a torus or a torus bundle over a circle. The key point of this paper is the observation of the relations between the sets of homotopy minimal periods of fibre-preserving maps and those of the induced maps on the base space, which is a torus or a circle. In fact, the sets of homotopy minimal periods are obvious for the circle, and were already found in [\textit{L. Alsedà, S. Baldwin, J. Llibre, R. Swanson} and \textit{W. Szlenk}, Pac. J. Math. 169, No. 1, 1--32 (1995; Zbl 0843.55004)] for the torus, where the concept of sets of homotopy minimal periods was called ``minimal sets of periods''.
scientific article; zbMATH DE number 5275717
scientific article