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

• Published in: Journal of fixed point theory and applications Vol. 21; no. 2; pp. 1 - 21
• Main Authors: Graff, Grzegorz, Lebiedź, Małgorzata, Myszkowski, Adrian
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01-06-2019; Springer Nature B.V
• Subjects: Analysis; Isomorphism; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Number theory; non-orientable compact surfaces; set of periods; Morse–Smale diffeomorphism; Lefschetz number; zeta function; periodic expansion; 37C25; 37E15; Primary 37D15
• ISSN: 1661-7738; 1661-7746
• DOI: 10.1007/s11784-019-0680-4

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
N
g
, 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
N
g
for any
g
.