Minimal number of periodic points for C 1 self-maps of compact simply-connected manifolds

Abstract

Let f be a self-map of a smooth compact connected and simplyconnected manifold of dimension m ≥ 3, r a fixed natural number. In this paper we define a topological invariant D m r [f ] which is the best lower bound for the number of r-periodic points for all C 1 maps homotopic to f . In case m = 3 we give the formula for D 3 r [f ] and calculate it for self-maps of S 2 × I.2000 Mathematics Subject Classification. Primary 37C25, 55M20 Secondary 37C05.

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

“…Thus, we can always represent basic sequences as a sum of one sequence of the type (A) and /2 − 1, DD 4 (1) sequences of the type (F). What is more, every DD 4 (1) sequence is also a DD (1) sequence for ≥ 4 [8]. We get finally, by the formula (9), that for any manifold of dimension ≥ 4,…”

“…Namely, let be a smooth self-map of a manifold M of dimension at least 4 and a fixed natural number. In the smooth homotopy class of one can create fixed points so that the sum of their indices of iterations is equal to the Lefschetz numbers of iterations and then remove all other -periodic points [8]. This strong result was obtained by the use of powerful Nielsen technics (Canceling and Creating Procedures proved by Jezierski in [16]).…”

“…MF ≤ ( ), obtained by taking continuous maps and homotopies in (2), always is ≤ 1. However, MF diff ≤ ( ) is usually greater than 1 [8]. In [13] we considered smooth self-maps with {L( )} In other words, J[ ] is equal to the minimal number of all periodic points with the periods less or equal to in a smooth homotopy class of for all sufficiently large .…”

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.…”

“…It would be difficult (also from the computational point of view) to follow the restrictions that come from the both conditions simultaneously, and thus we analyze the situation in which the fundamental group is trivial, so the Reidemeister relations disappear. Then the only obstacle that we have to control is related to the forms of local indices of iterations and we can apply the topological methods developed in [15] that were used to minimize the number of periodic points for C 1 self-maps of simply-connected manifold without boundary. As these technics work only in dimension at least 3 we have to assume that dim ∂ M ≥ 3.…”