A Note on the Periodic Structure of Transversal Maps on the Torus and Products of Spheres

Abstract

In the present article we study the periodic structure of some well-known classes of C 1 self-maps on the product of spheres of different dimensions: transversal maps, Morse-Smale diffeomorphisms and maps with all its periodic points hyperbolic. Our approach is via the Lefschetz fixed point theory. We give a complete characterization of the minimal set of Lefschetz periods for Morse-Smale diffeomorphisms on these spaces. We also consider C 1 maps with all its periodic points hyperbolic and we give conditions for these maps to have infinitely many periodic points. We describe the period set of the transversal maps on these spaces. Finally we applied these results to describe the periodic structure of similar classes of maps on compact, connected, simple connected and simple Lie groups.

Periodic structure of the transversal maps on surfaces

“…Theorems 1 and 2 are proved in sections 2 and 3, respectively. For obtaining these results we prove that (f m ) = 0 for m suciently large, the techniques used in order to get these results are based on the techniques developed in [23].…”

“…the homology spaces are either one-dimensional or trivial. Recently in [23] it was described the periodic structure of transversal self-maps on the n-torus, the product of spheres of the same dimension and on rational exterior spaces of a given rank.…”

Generalized Dold Sequences on Partially-Ordered Sets

“…it is realizable for the considered partial order, then by Theorem 27 it is a generalized Dold sequence. Furthermore, by (36):…”

“…Let us notice here, that Lefschetz numbers of iterations are very useful device in periodic point theory, cf. [1,4,9,18,23,36] and the references therein.…”