Periodic expansion in determining minimal sets of Lefschetz periods for Morse–Smale diffeomorphisms

Abstract

We apply the representation of Lefschetz numbers of iterates in the form of periodic expansion to determine the minimal sets of Lefschetz periods of Morse-Smale diffeomorphisms. Applying this approach we present an algorithmic method of finding the family of minimal sets of Lefschetz periods for Ng, a non-orientable compact surfaces without boundary of genus g. We also partially confirm the conjecture of Llibre and Sirvent (J Diff Equ Appl 19(3):402-417, 2013) proving that there are no algebraic obstacles in realizing any set of odd natural numbers as the minimal set of Lefschetz periods on Ng for any g.Mathematics Subject Classification. Primary 37D15, 37C25, 37E15.

Periods of Morse–Smale diffeomorphisms on $${\mathbb {S}}^n$$, $${\mathbb {S}}^m \times {\mathbb {S}}^n$$, $${\mathbb {C}}{\mathbf{P }}^n$$ and $${\mathbb {H}}{\mathbf{P }}^n$$

“…Without trying to be exhaustive, see for instance [5,19,[22][23][24]. This interest continues during this first part of the twenty-first century, see for example [2,3,9,10,[15][16][17][18].…”

Dold sequences, periodic points, and dynamics

“…Then The representation of a Dold sequence in the form (4) is called its periodic expansion, and is useful in many problems in periodic point theory. For example, in the periodic expansion of the Lefschetz numbers of iterations of Morse-Smale diffeomorphisms, each (reg k ) for odd k represents a periodic orbit of minimal period k. This observation allows one to easily determine the so-called minimal set of Lefschetz periods MPer L (f ) for Morse-Smale diffeomorphisms (see Graff et al [51]), studied by Guirao, Llibre, Sirvent, and other authors (we refer to [79] and the references therein for more on this).…”

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

“…The period set of Morse-Smale diffeomorphisms on S 2 , where studied before in [12]. See [11] for other developments in the subject of the minimal sets of Lefschetz periods.…”

“…In order to show that ℓ(f m ) are bounded away of zero, we need to compute the growth of the family of polynomials Q m (x). This is done in the following proposition: Note that if |a i | > 1 for all 1 ≤ i ≤ l, the equality (11) and Proposition 11 yields: there exist positive constants C 1 , . .…”