Periods of self-maps on S2 via their homology
DOI:
https://doi.org/10.3842/umzh.v76i1.7668Keywords:
Self-maps of the 2-dimensional sphere, set of periods, periodic points, Lefschetz numbersAbstract
UDC 517.9
As usual, we denote а 2-dimensional sphere by S2. We study the periods of periodic orbits of the maps f:S2→S2 that are either continuous or C1 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 S2 together. We note that every time when a map f:S2→S2 increases its structure, the number of periodic orbits provided by its action on the homology increases.
References
I. N. Baker, Fix points of polynomials and rational functions, J. London Math. Soc., 39, 615–622 (1964). DOI: https://doi.org/10.1112/jlms/s1-39.1.615
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). DOI: https://doi.org/10.1090/S0002-9947-00-02608-8
J. L. Garcia-Guirao, J. Llibre, C1 self-maps on Sn, Sn×Sm, CPn and HPn with all their periodic orbits hyperbolic, Taiwanese J. Math., 16, 323–334 (2012). DOI: https://doi.org/10.11650/twjm/1500406543
J. L. Garcia-Guirao, J. Llibre, Periodic structure of fransversal maps on CPn, HPn and Sp×Sq, Qual. Theory Dyn. Syst., 12, 417–425 (2013). DOI: https://doi.org/10.1007/s12346-013-0099-z
J. L. Garcia-Guirao, J. Llibre, Periods of continuous maps on some compact spaces, J. Difference Equat. and Appl., 23, 1–7 (2017). DOI: https://doi.org/10.1080/10236198.2017.1304932
T. Y. Li, J. A. Yorke, Period three implies chaos, Amer. Math. Monthly, 82, № 10, 985–992 (1975). DOI: https://doi.org/10.1080/00029890.1975.11994008
J. Llibre, Lefschetz numbers for periodic points, Contemp. Math., 152, 215–227 (1993). DOI: https://doi.org/10.1090/conm/152/01325
J. Llibre, A note on the set of periods of transversal homological sphere self-maps, J. Difference Equat. and Appl., 9, 417–422 (2003). DOI: https://doi.org/10.1080/1023619021000047833
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). DOI: https://doi.org/10.1007/978-1-4612-0881-5
