Every finite set of natural numbers is realizable as algebraic periods of a Morse$\unicode{x2013}$Smale diffeomorphism

TL;DR

This paper proves that any finite set of natural numbers can be realized as the algebraic periods of a Morse–Smale diffeomorphism on surfaces, solving an open problem in dynamical systems.

Contribution

It demonstrates the existence of Morse–Smale diffeomorphisms with prescribed algebraic periods on both orientable and non-orientable surfaces, and provides bounds on conjugacy classes.

Findings

Any finite subset of natural numbers can be realized as algebraic periods.

Constructs Morse–Smale diffeomorphisms with prescribed algebraic periods.

Estimates the number of conjugacy classes of such diffeomorphisms.

Abstract

A given self-map $f\colon M\to M$ of a compact manifold determines the sequence $(L(f^n))$ of the Lefschetz numbers of its iterations. We consider its dual sequence $(a_n(f))$ given by the M\"obius inversion formula. The set ${\mathcal AP}(f)=\{ n\in \mathbb N\ \colon\ a_n(f)\neq 0\}$ is called the set of algebraic periods. We solve an open problem existing in literature by showing that for every finite subset ${\mathcal A}$ of natural numbers there exist an orientable surface $S_{\rm g}$, as well as a non-orientable surface $N_{\rm g}$, of genus ${\rm g}$, and a Morse$\unicode{x2013}$Smale diffeomorphism $f$ of this surface such that $\mathcal{AP}(f)=\mathcal{A}$. For such a map it implies the existence of points of a minimal period $n$ for each odd $n \in \mathcal{A}$. For the orientation-reversing Morse$\unicode{x2013}$Smale diffeomorphisms of $S_{\rm g}$, we identify strong restrictions on ${\mathcal AP}(f)$. Our method also provides an estimate of the number of conjugacy classes of mapping classes containing Morse$\unicode{x2013}$Smale diffeomorphisms, which is exponential in ${\rm g}$.