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

Abstract

Abstract. We study the minimal set of (Lefschetz) periods of the C 1 MorseSmale diffeomorphisms on a non-orientable compact surface without boundary inside its class of homology. In fact our study extends to the C 1 diffeomorphisms on these surfaces having finitely many periodic orbits all of them hyperbolic and with the same action on the homology as the Morse-Smale diffeomorphisms.We mainly have two kind of results. First we completely characterize the minimal sets of periods for the C 1 Morse-Smale diffeomorphisms on non-orientable compact surface without boundary of genus g with 1 ≤ g ≤ 9. But the proof of these results provides an algorithm for characterizing these minimal sets of periods for the C 1 Morse-Smale diffeomorphisms on non-orientable compact surfaces without boundary of arbitrary genus.Second we study what kind of subsets of positive integers can be minimal sets of periods of the C 1 Morse-Smale diffeomorphisms on a non-orientable compact surface without boundary.

C^1 self-maps on closed manifolds with finitely many periodic points all of them hyperbolic

“…We remark that Theorem 1.1 allows to weaken the hypothesis of the results of [2], [7], [9], [10]. In these articles the periodic orbits of Morse-Smale diffeomorphisms on n-dimensional torus, orientable and non-orientable surfaces are studied.…”

Cyclotomic polynomials and minimal sets of Lefschetz periods

“…In relation to Question 1, all possible L z are listed for g ¼ 0; 1; 2; 3, in [13,14]. Using the properties of the cyclotomic polynomials listed in Section 3, we have these easy results (cf.…”

“…Using the properties of the cyclotomic polynomials listed in Section 3, we have these easy results (cf. [13,14]):…”

“…In particular, they play an important rôle in the study of the set of periods for a class of differentiable maps on manifolds, see [4,5,8,9,10,13] and references within. In [13], it was shown that the description of the minimal set of Lefschetz periods is equivalent to a problem of factorization of the product of cyclotomic polynomials. In the present paper, we raise this question and give partial but extensive answers.…”

“…In [13], it was shown that the set L z is the minimal set of Lefschetz periods of a MorseSmale diffeomorphism f on an orientable surface of genus g, whose induced map in 1th rational homology group has qðtÞ as the characteristic polynomial. The study of the set L z is fundamental in understanding the periodic structure of the map f. In this paper, we shall call the set L z the minimal set of Lefschetz periods associated with qðtÞ.…”