Periods of self-maps on $\rm S^2$ via their homology

Jaume Llibre

doi:10.3842/umzh.v76i1.7668

Jaume Llibre Departament de Matemàtiques, Universitat Autònoma de Barcelona, Spain

DOI: https://doi.org/10.3842/umzh.v76i1.7668

Keywords: Self-maps of the 2-dimensional sphere, set of periods, periodic points, Lefschetz numbers

Abstract

UDC 517.9

As usual, we denote а $2$-dimensional sphere by $\rm S^2$. We study the periods of periodic orbits of the maps $f\colon \rm S^2 \rightarrow \rm S^2$ that are either continuous or $C^1$ with all their periodic orbits being hyperbolic, or transverse, or holomorphic, or transverse holomorphic. For the first time, we summarize all known results on the periodic orbits of these distinct kinds of self-maps on $\rm S^2$ together. We note that every time when a map $f\colon \rm S^2 \rightarrow \rm S^2$ increases its structure, the number of periodic orbits provided by its action on the homology increases.

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