Combinatorial scheme of finding minimal number of periodic points for smooth self-maps of simply connected manifolds

Abstract

Abstract. Let M be a closed smooth connected and simply connected manifold of dimension m at least 3, and let r be a fixed natural number. The topological invariant D Mathematics Subject Classification (2010). Primary 37C25, 55M20; Secondary 37C05.

Minimal number of periodic points of smooth boundary-preserving self-maps of simply-connected manifolds

“…4 in [17], see also Lemma 4.8 in [15]). This statement is true for each r in dimension at least 4 and for odd r also in dimension 3.…”

Dold sequences, periodic points, and dynamics

“…The determination of the exact form of possible indices of iterations for smooth maps (see [50]) turned out to have many topological consequences. In particular, based on that result Graff and Jezierski constructed a smooth branch of Nielsen periodic point theory, obtaining invariants that allow the minimal number of periodic points in a smooth homotopy class to be computed [47][48][49].…”

Minimization of the number of periodic points for smooth self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers

“…This strong result was obtained by the use of powerful Nielsen technics (Canceling and Creating Procedures proved by Jezierski in [16]). On the other hand, the created fixed points have indices that are DD (1) sequences and it is known that MF diff ≤ ( ) can be realized by a map with all -periodic points being fixed points [11]. Thus, the minimal number of -periodic points in the smooth homotopy class of is given by item IV.…”

“…Let be a prime number, observe that α ∈ B =⇒ α ≤ 2 (11) Indeed, by the equality √ ≤ ( ) which holds for > 6, cf. [21], we get that if ∈ B, then √ ≤ ( ) ≤ , thus ≤ 2 .…”