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On the growth of the number of periodic points for smooth self-maps of a compact manifold
Authors:
Grzegorz Graff and Jerzy Jezierski
Journal:
Proc. Amer. Math. Soc. 135 (2007), 3249-3254
MSC (2000):
Primary 37C25, 55M20; Secondary 37C05
Posted:
June 20, 2007
MathSciNet review:
2322756
Full-text PDF Free Access
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Additional Information
Abstract: Let  be a continuous self-map of a smooth compact connected and simply-connected manifold of dimension  . We show that in the homotopy class of  there is a  map with less then  periodic points, up to any given fixed period  .
- [BaBo]
I.
K. Babenko and S.
A. Bogatyĭ, Behavior of the index of periodic points under
iterations of a mapping, Izv. Akad. Nauk SSSR Ser. Mat.
55 (1991), no. 1, 3–31 (Russian); English
transl., Math. USSR-Izv. 38 (1992), no. 1,
1–26. MR
1130026 (93a:58139)
- [B]
Robert
F. Brown, The Lefschetz fixed point theorem, Scott, Foresman
and Co., Glenview, Ill.-London, 1971. MR 0283793
(44 #1023)
- [CMPY]
S.-N. Chow, J. Mallet-Paret, 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.
- [D1]
A.
Dold, Fixed point indices of iterated maps, Invent. Math.
74 (1983), no. 3, 419–435. MR 724012
(85c:54077), http://dx.doi.org/10.1007/BF01394243
- [D2]
Albrecht
Dold, Lectures on algebraic topology, Classics in Mathematics,
Springer-Verlag, Berlin, 1995. Reprint of the 1972 edition. MR 1335915
(96c:55001)
- [GNP]
G. Graff and M. Nowak-Przygodzki, Fixed point indices of iterations of
 maps in  , Discrete Cont. Dyn. Syst. 16 (2006), 843-856.
- [Je]
Jerzy
Jezierski, Wecken’s theorem for periodic points in dimension
at least 3, Topology Appl. 153 (2006), no. 11,
1825–1837. MR 2227029
(2006m:37025), http://dx.doi.org/10.1016/j.topol.2005.06.008
- [JM]
Jerzy
Jezierski and Wacław
Marzantowicz, Homotopy methods in topological fixed and periodic
points theory, Topological Fixed Point Theory and Its Applications,
vol. 3, Springer, Dordrecht, 2006. MR 2189944
(2006i:55003)
- [K]
Vadim
Yu. Kaloshin, Generic diffeomorphisms with superexponential growth
of number of periodic orbits, Comm. Math. Phys. 211
(2000), no. 1, 253–271. MR 1757015
(2001e:37035), http://dx.doi.org/10.1007/s002200050811
- [SS]
M.
Shub and D.
Sullivan, A remark on the Lefschetz fixed point formula for
differentiable maps, Topology 13 (1974),
189–191. MR 0350782
(50 #3274)
- [BaBo]
- I. K. Babenko, S. A. Bogatyi, Behavior of the index of periodic points under iterations of a mapping, Math. USSR Izv. 38 (1992), 1-26. MR 1130026 (93a:58139)
- [B]
- R. F. Brown, The Lefschetz Fixed Point Theorem, Glenview, New York, 1971.MR 0283793 (44:1023)
- [CMPY]
- S.-N. Chow, J. Mallet-Paret, 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.
- [D1]
- A. Dold, Fixed point indices of iterated maps, Invent. Math. 74 (1983), 419-435.MR 0724012 (85c:54077)
- [D2]
- A. Dold, Lectures on algebraic topology, Springer-Verlag, Berlin, 1995.MR 1335915 (96c:55001)
- [GNP]
- G. Graff and M. Nowak-Przygodzki, Fixed point indices of iterations of maps in , Discrete Cont. Dyn. Syst. 16 (2006), 843-856.
- [Je]
- J. Jezierski, Wecken's theorem for periodic points in dimension at least
 , Topology Appl. 153 (2006), no. 11, 1825-1837.MR 2227029
- [JM]
- 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, 2005.MR 2189944 (2006i:55003)
- [K]
- V. Kaloshin, Generic diffeomorphisms with superexponential growth of number of periodic orbits, Comm. Math. Phys. 211 (2000), no. 1, 253-271.MR 1757015 (2001e:37035)
- [SS]
- M. Shub, P. Sullivan, A remark on the Lefschetz fixed point formula for differentialble maps, Topology 13 (1974), 189-191.MR 0350782 (50:3274)
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Additional Information
Grzegorz Graff
Affiliation:
Faculty of Applied Physics and Mathematics, Gdansk University of Technology, Narutowicza 11/12, 80-952 Gdansk, Poland
Email:
graff@mif.pg.gda.pl
Jerzy Jezierski
Affiliation:
Institute of Applications of Mathematics, Warsaw University of Life Sciences (SGGW), Nowoursynowska 159, 00-757 Warsaw, Poland
Email:
jezierski@acn.waw.pl
DOI:
http://dx.doi.org/10.1090/S0002-9939-07-08836-3
PII:
S 0002-9939(07)08836-3
Keywords:
Periodic points,
$C^1$ maps,
indices of iterations,
Nielsen number.
Received by editor(s):
March 30, 2006
Received by editor(s) in revised form:
June 30, 2006
Posted:
June 20, 2007
Additional Notes:
This research was supported by KBN grant No 1 P03A 03929.
Communicated by:
Jane M. Hawkins
Article copyright:
© Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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