Journal of the Mathematical Society of Japan

Homotopy minimal periods for expanding maps on infra-nilmanifolds

Jong Bum LEE and Xuezhi ZHAO

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We prove that the sets of homotopy minimal periods for expanding maps on n -dimensional infra-nilmanifolds are uniformly cofinite,i.e., there exists a positive integer m 0 , which depends only on n , such that for any integer m m 0 , for any n -dimensional infra-nilmanifold M , and for any expanding map f on M , any self-map on M homotopic to f has a periodic point of least period m , namely, [ m 0 , ) H P e r ( f ) . This extends the main result, Theorem 4.6, of [13] from periods to homotopy periods.

Article information

J. Math. Soc. Japan, Volume 59, Number 1 (2007), 179-184.

First available in Project Euclid: 25 May 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55M20: Fixed points and coincidences [See also 54H25]
Secondary: 57S30: Discontinuous groups of transformations

essentially reducible expanding maps homotopy minimal periods infra-nilmanifolds


LEE, Jong Bum; ZHAO, Xuezhi. Homotopy minimal periods for expanding maps on infra-nilmanifolds. J. Math. Soc. Japan 59 (2007), no. 1, 179--184. doi:10.2969/jmsj/1180135506.

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  • L. Alsedà, S. Baldwin, J. Llibre, R. Swanson and W. Szlenk, Minimal sets of periods for torus maps via Nielsen numbers, Pacific J. Math., 169 (1995), 1–32.
  • K. Dekimpe and K. B. Lee, Expanding maps on infra-nilmanifolds of homogeneous type, Trans. Amer. Math. Soc., 355 (2003), 1067–1077.
  • K. Dekimpe and K. B. Lee, Expanding maps, Anosov diffeomorphisms and affine structures on infra-nilmanifolds, Topology Appl., 130 (2003), 259–269.
  • J. L. Dyer, A nilpotent Lie algebra with nilpotent automorphism group, Bull. Amer. Math. Soc., 76 (1970), 52–56.
  • D. Epstein and M. Shub, Expanding endomorphisms of flat manifolds, Topology, 7 (1968), 139–141.
  • M. Gromov, Groups of polynomial growth and expanding maps, Publ. Math. Inst. Hautes Etudes Sci., 53 (1981), 53–73.
  • B. Halpern, Periodic points on tori, Pacific J. Math., 83 (1979), 117–133.
  • P. Heath and E. Keppelmann, Fibre techniques in Nielsen periodic point theory on nil and solvmanifolds I, Topology Appl., 76 (1997), 217–247.
  • J. Jezierski, J. Kędra and W. Marzantowicz, Homotopy minimal periods for NR-solvmanifolds maps, Topology Appl., 144 (2004), 29–49.
  • S. Kwasik and K. B. Lee, The Nielsen numbers of homotopically periodic maps of infra-nilmanifolds, J. London Math. Soc. (2), 38 (1988), 544–554.
  • S. W. Kim, J. B. Lee and K. B. Lee, Averaging formula for Nielsen numbers, Nagoya Math. J., 178 (2005), 37–53.
  • H. Lee and K. B. Lee, Expanding maps on $2$-step infra-nilmanifolds, Topology Appl., 117 (2002), 45–58.
  • J. B. Lee and K. B. Lee, Lefschetz numbers for continuous maps, and periods for expanding maps on infra-nilmanifolds, J. Geom. Phys., 56 (2006), 2011–2023.
  • K. B. Lee and F. Raymond, Rigidity of almost crystallographic groups, Contemporary Math. Amer. Math. Soc., 44 (1985), 73–78.
  • M. Shub, Endomorphism of compact differentiable manifolds, Amer. J. Math., 157 (1993), 87–93.