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

  • Jaume Llibre Departament de Matemàtiques, Universitat Autònoma de Barcelona, Spain
Keywords: Self-maps of the 2-dimensional sphere, set of periods, periodic points, Lefschetz numbers


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.


I. N. Baker, Fix points of polynomials and rational functions, J. London Math. Soc., 39, 615–622 (1964).

R. F. Brown, The Lefschetz fixed point theorem, Scott, Foresman and Company, Glenview, IL (1971).

N. Fagella, J. Llibre, Periodic points of holomorphic maps via Lefstchetz numbers, Trans. Amer. Math. Soc., 352, 4711–4730 (2000).

J. L. Garcia-Guirao, J. Llibre, $C^{1}$ self-maps on $S^{n},$ $S^{n} × S^{m},$ $C$P$^{n}$ and $HP^{n}$ with all their periodic orbits hyperbolic, Taiwanese J. Math., 16, 323–334 (2012).

J. L. Garcia-Guirao, J. Llibre, Periodic structure of fransversal maps on $CP^n,$ $HP^n$ and $S^p × S^q$, Qual. Theory Dyn. Syst., 12, 417–425 (2013).

J. L. Garcia-Guirao, J. Llibre, Periods of continuous maps on some compact spaces, J. Difference Equat. and Appl., 23, 1–7 (2017).

T. Y. Li, J. A. Yorke, Period three implies chaos, Amer. Math. Monthly, 82, № 10, 985–992 (1975).

J. Llibre, Lefschetz numbers for periodic points, Contemp. Math., 152, 215–227 (1993).

J. Llibre, A note on the set of periods of transversal homological sphere self-maps, J. Difference Equat. and Appl., 9, 417–422 (2003).

A. N. Sharkovsky, Coexistence of the cycles of a continuous mapping of the line into itself} (in Russian), Ukr. Math. Zh., 16, № 1, 61–71 (1964).

J. W. Vick, Homology theory, 2nd ed., Springer-Verlag (1994).

How to Cite
Llibre, J. “Periods of Self-Maps on $\rm S^2$ via Their Homology”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 76, no. 1, Feb. 2024, pp. 72 -74, doi:10.3842/umzh.v76i1.7668.
Research articles