Abstract.
A well-known example, given by Shub, shows that for any |d| ≥ 2 there is a self-map of the sphere Sn, n ≥ 2, of degree d for which the set of non-wandering points consists of two points. It is natural to ask which additional assumptions guarantee an infinite number of periodic points of such a map. In this paper we show that if a continuous map f : Sn → Sn commutes with a free homeomorphism g : Sn → Sn of a finite order, then f has infinitely many minimal periods, and consequently infinitely many periodic points. In other words the assumption of the symmetry of f originates a kind of chaos. We also give an estimate of the number of periodic points.
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*Research supported by KBN grant nr 2 P03A 045 22.
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Jezierski, J., Marzantowicz, W. A symmetry of sphere map implies its chaos*. Bull Braz Math Soc, New Series 36, 205–224 (2005). https://doi.org/10.1007/s00574-005-0037-z
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DOI: https://doi.org/10.1007/s00574-005-0037-z