The number of periodic points of smooth maps

Abstract

Abstract. Let/: M -* M be a C 1 map on a compact manifold. We give a topological condition under which / has an even number of periodic points with a given period.We also obtain a sufficient condition, in terms of homology, for / to have infinitely many periodic points. In this paper, we give two applications of the Dold's equalities to the study of periodic points of smooth maps. The first application is concerned with the problem of whether the number v(n) of periodic points of a given period n is even or odd. We give a condition on n and on the homotopy class of the map under which v{n) is even. This generalizes results in [14,15,18].In the second application, we generalize some results of Franks [8,9] on the existence of infinitely many periodic points.In the last section, we generalize the Dold's equalities to isolated sets of periodic points.

Periodic Structure of Transversal Maps on $$\mathbb{C }$$ P $$^{n}$$ , $$\mathbb{H }$$ P $$^{n}$$ and $$\mathbb{S }^{p}\times \mathbb{S }^{q}$$

“…Perhaps the best known example in this direction are the results contained in the seminal paper entitle Period three implies chaos for continuous self-maps on the interval, see [11]. In the case of transversal maps different studies appear in the literature in general from other point of view, see for instance Franks [7,8]; Matsuoka [15]; Babenko y Bogatyi [1]; Casasayas, Llibre and Nuñes [3]; Llibre, Paraños and Rodriguez [12], Llibre and Swanson [14] and Fagella and Llibre [6].…”

The Nielsen and Reidemeister theories of iterations on infra-solvmanifolds of type (𝑅) and poly-Bieberbach groups

“…The study of periodic points by using the Lefschetz theory has been done extensively by many authors in the literature such as [27], [9], [2], [24], [41]. A natural question is to ask how much information we can get about the set of essential periodic points of f or about the set of (homotopy) minimal periods of f from the study of the sequence {N (f k )} of the Nielsen numbers of iterations of f .…”

Minimal sets of periods for Morse–Smale diffeomorphisms on non-orientable compact surfaces without boundary

“…Probably for continuous self-maps on compact manifolds the most useful tool for studying the existence of fixed and periodic points is the Lefschetz Fixed Point Theorem and its improvements, see for instance [1,2,6,7,8,10,15,16,20,23]. The Lefschetz zeta function ζ f (t) simplifies the study of the periodic points of f .…”