The least number of n-periodic points on tori can be realized by a smooth map

Jerzy Jezierski

DOI: http://dx.doi.org/10.12775/TMNA.2015.084

Abstract


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.

Keywords


Fixed point; periodic point; Nielsen fixed point theory; Dold congruences; least number of periodic points

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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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