The least number of n-periodic points on tori can be realized by a smooth map
DOI:
https://doi.org/10.12775/TMNA.2015.084Keywords
Fixed point, periodic point, Nielsen fixed point theory, Dold congruences, least number of periodic pointsAbstract
We give an algebraic proof of the Theorem of Cheng Ye You that the least number of $n$-periodic points, in the continuous homotopy class of a self-map of a~torus, can be realized by a smooth map.References
I.K. Babenko and S.A. Bogatyi, The behavior of the index of periodic points under iterations of a mapping, Math. USSR Izv. 38 (1992), 1-26.
Sc R. Brooks, R. Brown, J. Pak and D. Taylor, Nielsen numbers of maps of tori, Proc. Amer. Math. Soc. 52 (1975), 398-400.
R. F. Brown, The Lefschetz Fixed Point Theorem, Glenview, New York, 1971.
S.N. Chow, J. Mallet-Paret and J.A. Yorke, A periodic point index which is a bifurcation invariant, Geometric dynamics (Rio de Janeiro, 1981), 109-131, Springer Lecture Notes in Math. 1007, Berlin 1983.
A. Dold, Fixed point indices of iterated maps, Invent. Math. 74 (1983), 419-435.
G. Graff and J. Jezierski, Minimal number of periodic points for C1 self-maps of compact simply-connected manifolds, Forum Math. 21 (2009), no. 3, 491-509.
G. Graff and J. Jezierski, Minimizing the number of periodic points for smooth maps. Non-simply connected case, Topology Appl. 158 (2011), no. 3, 276-290.
G. Graff, J. Jezierski and P. Nowak-Przygodzki, Fixed point indices of iterated smooth maps in arbitrary dimension J. Differential Equations 251 (2011), no. 6, 1526-1548.
G. Graff and P. Nowak-Przygodzki, Fixed point indices of iterations of C1 maps in R^3, Discrete Cont. Dyn. Systems 16 (2006), no. 4, 843-856.
B. Halpern, Periodic points on tori, Pacific J. Math. 83 (1979), no. 1, 117-133.
Ph. Heath and E. Keppelmann, Fibre techniques in Nielsen periodic point theory on nil and solvmanifolds I, Topology Appl. 76 (1997), no. 3, 217-247.
Ph. Heath and E. Keppelmann, Fibre techniques in Nielsen periodic point theory on nil and solvmanifolds II, Topology Appl. 106 (2000), no. 2, 149-167.
J. Jezierski, Wecken's theorem for periodic points in dimension at least 3, Topology Appl. 153 (2006), no. 11, 1825-1837.
J. Jezierski, The least number, of n -periodic points of a self-map of a solvmanifold, can be realised by a smooth map, Topology Appl. 158 (2011), no. 9, 1113-1120.
J. Jezierski, Least number of periodic points of self-maps of Lie groups, Acta Math. Sinica 30 (2014), no. 9, 1477-1494.
J. Jezierski and W. Marzantowicz, Homotopy methods in topological fixed and periodic points theory, Topological Fixed Point Theory and Its Applications, Vol. 3. Springer, Dordrecht, 2006. xii+319 pp.
B.J. Jiang, Lectures on the Nielsen Fixed Point Theory, Contemp. Math. 14, Amer. Math. Soc., Providence 1983.
B.J. Jiang, Fixed point classes from a differential viewpoint, in: Lecture Notes in Math. 886, Springer, (1981), 163-170.
M. Shub and P. Sullivan, A remark on the Lefschetz fixed point formula for differentiable maps, Topology 13 (1974), 189-191.
C.Y. You, The least number of periodic points on tori Adv. in Math. (China) 24 (1995), no. 2, 155-160.
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