Abstract
In this paper, we show that the Nielsen number N(f) of any map f on an infra-nilmanifold is either equal to |L(f)|, where L(f) is the Lefschetz number of that map, or equal to|L(f)−L(f +)|, where f + is a lift of f to a 2-fold covering of that infra-nilmanifold. By exploiting the exact nature of this relationship for all powers of f, we prove that the Nielsen dynamical zeta function for a map on an infra-nilmanifold is always a rational function.
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References
D. Anosov, The Nielsen numbers of maps of nilmanifolds. Uspekhi. Mat. Nauk 40 (1985), 133–134 (in Russian); English transl.: Russian Math. Surveys 40 (1985), 149–150.
R. F. Brown, The Lefschetz Fixed Point Theorem. Scott, Foresman and Co., Glenview, Ill., 1971.
Dekimpe K., De Rock B., Malfait W.: The Anosov theorem for flat generalized Hantzsche-Wendt manifolds. J. Geom. Phys. 52, 174–185 (2004)
Dekimpe K., De Rock B., Malfait W.: The Anosov relation for Nielsen numbers of maps of infra-nilmanifolds. Monatsh. Math. 150, 1–10 (2007)
Dekimpe K., De Rock B., Penninckx P.: Anosov theorem for infranilmanifolds with a 2-perfect holonomy group. Asian J. Math. 15, 539–548 (2011)
E. Fadell and S. Husseini, On a theorem of Anosov on Nielsen numbers for nilmanifolds. In: Nonlinear Functional Analysis and Its Applications (Maratea, 1985), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 173, Reidel, Dordrecht, 1986, 47–53.
A. Fel’shtyn, Dynamical zeta functions, Nielsen theory and Reidemeister torsion. Mem. Amer. Math. Soc. 147 (2000), xii+146.
A. L. Fel’shtyn, New zeta functions for dynamical systems and Nielsen fixed point theory. In: Topology and Geometry—Rohlin Seminar, Lecture Notes in Math. 1346, Springer, Berlin, 1988, 33–55.
Fel’shtyn A.: Nielsen zeta function, 3-manifolds, and asymptotic expansions in Nielsen theory. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 266, 312–329 (2000)
A. Fel’shtyn and R. Hill, Dynamical zeta functions, congruences in Nielsen theory and Reidemeister torsion. In: Nielsen Theory and Reidemeister Torsion (Warsaw, 1996), Banach Center Publ. 49, Polish Acad. Sci., Warsaw, 1999, 77–116.
Ha K.Y., Lee J.B., Penninckx P.: Anosov theorem for coincidences on special solvmanifolds of type (R). Proc. Amer. Math. Soc. 139, 2239–2248 (2011)
Keppelmann E.C., McCord C.K.: The Anosov theorem for exponential solvmanifolds. Pacific J. Math. 170, 143–159 (1995)
Lee J. B., Lee K. B.: Averaging formula for Nielsen numbers of maps on infra-solvmanifolds of type (R). Nagoya Math. J. 196, 117–134 (2009)
Lee K.B.: Maps on infra-nilmanifolds. Pacific J. Math. 168, 157–166 (1995)
Li L.: On the rationality of the Nielsen zeta function. Adv. Math. (China) 23, 251–256 (1994)
Pilyugina V.B., Fel’shtyn A.L.: The Nielsen zeta function. Funktsional. Anal. i Prilozhen. 19, 61–67 (1985)
Smale S.: Differentiable dynamical systems. Bull. Amer. Math. Soc. 73, 747–817 (1967)
Wong P.: Reidemeister zeta function for group extensions. J. Korean Math. Soc. 38, 1107–1116 (2001)
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Dekimpe, K., Dugardein, GJ. Nielsen zeta functions for maps on infra-nilmanifolds are rational. J. Fixed Point Theory Appl. 17, 355–370 (2015). https://doi.org/10.1007/s11784-014-0165-4
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DOI: https://doi.org/10.1007/s11784-014-0165-4