Homotopy minimal periods for maps of three dimensional nilmanifolds

Minimal periods of holomorphic maps on complex tori

“…To avoid this difficulty many authors studied the set of homotopy minimal periods, i.e. minimal periods which are preserved by any homotopy (see for instance [1], [10], [7], and [8, Chapt. VI] for an exposition of known results).…”

Nielsen type numbers and homotopy minimal periods for maps on 3–solvmanifolds

“…Since the homotopy minimal period is preserved under a small perturbation of a self-map f on a manifold X , we can say that the set HPer.f / of homotopy minimal periods of f describes the rigid part of the dynamics of f . A complete description of the set of homotopy minimal periods of all self-maps was obtained on 3-nilmanifolds by Jezierski and Marzantowicz [6], and Lee and Zhao [10] and on 3-solvmanifolds by Jezierski, Kȩdra and Marzantowicz [5].…”

“…Jezierski and Marzantowicz in [6], and the authors in [10] studied homotopy minimal periods for maps on 3-manifolds with Nil-geometry. In [5] (see also Kim, Lee and Yoo [9]), Jezierski, Kȩdra and Marzantowicz carried out a further study of homotopy minimal periods for maps on 3-solvmanifolds.…”

Nielsen type numbers and homotopy minimal periods for maps on the 3-nilmanifolds

“…Since the homotopy minimal period is preserved under a small perturbation of a self-map f on a manifold X, we can say that the set HPer(f ) of homotopy minimal periods of f describes the rigid part of dynamics of f . Jezierski and Marzantowicz in [4] gave a description of the sets of homotopy minimal periods of all self-maps on the 3-nilmanifolds. They used the facts that every 3-nilmanifold M admits a fibration with S 1 as the fiber and T 2 as the base and that every map on M is homotopic to a fiber preserving map of this fibration, so that they could use the previous results on S 1 and T 2 .…”