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When a smooth self-map of a semi-simple Lie group can realize the least number of periodic points

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Abstract

There are two algebraic lower bounds of the number of n-periodic points of a self-map f: M → M of a compact smooth manifold of dimension at least 3: NF n(f) = min{#Fix(g n); g ~ f; g continuous} and NJD n(f) = min{#Fix(g n); g ~ f; g smooth}. In general, NJD n(f) may be much greater than NF n(f). We show that for a self-map of a semi-simple Lie group, inducing the identity fundamental group homomorphism, the equality NF n(f) = NJD n(f) holds for all n ⇔ all eigenvalues of a quotient cohomology homomorphism induced by f have moduli ≤ 1.

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Acknowledgements

This work was supported by the National Science Center, Poland (Grant No. UMO- 2014/15/B/ST1/01710).

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  1. Institute of Applications of Informatics and Mathematics, Warsaw University of Life Sciences (SGGW), Warsaw, 00-757, Poland

    Jerzy Jezierski

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  1. Jerzy Jezierski
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Correspondence to Jerzy Jezierski.

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Dedicated to Professor Boju Jiang on the Occasion of His 80th Birthday

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Jezierski, J. When a smooth self-map of a semi-simple Lie group can realize the least number of periodic points. Sci. China Math. 60, 1579–1590 (2017). https://doi.org/10.1007/s11425-016-9099-9

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