The generalized Lefschetz number of homeomorphisms on punctured disks
Takashi MATSUOKA
Source: J. Math. Soc. Japan Volume 61, Number 4
(2009), 1205-1241.
Abstract
We compute the generalized Lefschetz number of orientation-preserving self-homeomorphisms of a compact punctured disk, using the fact that homotopy classes of these homeomorphisms can be identified with braids. This result is applied to study Nielsen-Thurston canonical homeomorphisms on a punctured disk. We determine, for a certain class of braids, the rotation number of the corresponding canonical homeomorphisms on the outer boundary circle. As a consequence of this result on the rotation number, it is shown that the canonical homeomorphisms corresponding to some braids are pseudo-Anosov with associated foliations having no interior singularities.
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Keywords: generalized Lefschetz number; fixed point; periodic point; braid; Nielsen-Thurston classification theory of homeomorphisms; punctured disk
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Permanent link to this document: http://projecteuclid.org/euclid.jmsj/1257520505
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Zentralblatt MATH identifier: 05651149
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References
D. Benardete, M. Gutierrez and Z. Nitecki, Braids and the Nielsen-Thurston classification, J. Knot Theory Ramifications, 4 (1995), 549–618.
Mathematical Reviews (MathSciNet): MR1361083
Zentralblatt MATH: 0874.57010
Digital Object Identifier: doi:10.1142/S0218216595000259
M. Bestvina and M. Handel, Train-tracks for surface homeomorphisms, Topology, 34 (1995), 109–140.
Mathematical Reviews (MathSciNet): MR1308491
Digital Object Identifier: doi:10.1016/0040-9383(94)E0009-9
J. S. Birman, Braids, Links, and Mapping Class Groups, Ann. Math. Studies, 82, Princeton Univ. Press, Princeton, 1974.
Mathematical Reviews (MathSciNet): MR375281
Zentralblatt MATH: 0305.57013
P. Boyland, Braid types and a topological method of proving positive entropy (1984), unpublished.
P. Boyland, Topological methods in surface dynamics, Topology Appl., 58 (1994), 223–298.
Mathematical Reviews (MathSciNet): MR1288300
Zentralblatt MATH: 0810.54031
Digital Object Identifier: doi:10.1016/0166-8641(94)00147-2
R. F. Brown, The Lefschetz Fixed Point Theorem, Scott-Foresman, Glenview, 1971.
Mathematical Reviews (MathSciNet): MR283793
E. Fadell and S. Husseini, The Nielsen number on surfaces, Topological Methods in Nonlinear Functional Analysis (eds. S. P. Singh et al.), Contemp. Math., 21, Amer. Math. Soc., Providence, 1983, pp. 59–98.
Mathematical Reviews (MathSciNet): MR729505
Zentralblatt MATH: 0563.55001
A. Fathi, F. Laudenbach, and V. Poénaru, Travaux de Thurston sur les surfaces, Astérisque, 66–67, Soc. Math. France, Paris, 1979.
Mathematical Reviews (MathSciNet): MR568308
J. Franks and M. Misiurewicz, Cycles for disk homeomorphisms and thick trees, Nielsen Theory and Dynamical Systems (ed. C. McCord), Contemp Math., 152, Amer. Math. Soc., Providence, 1993, pp. 69–139.
Mathematical Reviews (MathSciNet): MR1243471
Zentralblatt MATH: 0793.58029
J. Guaschi, Nielsen theory, braids and fixed points of surface homeomorphisms, Top. Appl., 117 (2002), 199–230.
Mathematical Reviews (MathSciNet): MR1875911
Zentralblatt MATH: 1001.54027
Digital Object Identifier: doi:10.1016/S0166-8641(01)00019-0
H. Hamidi-Tehrani and Z. Chen, Surface diffeomorphisms via train-tracks, Top. Appl., 73 (1996), 141–167.
Mathematical Reviews (MathSciNet): MR1416757
Zentralblatt MATH: 0870.57048
Digital Object Identifier: doi:10.1016/0166-8641(96)00024-7
H.-H. Huang and B. Jiang, Braids and periodic solutions, Topological Fixed Point Theory and Applications (ed. B. Jiang), Lecture Notes in Math., 1411, Springer-Verlag, Berlin, Heidelberg, New York, 1989, pp. 107–123.
Mathematical Reviews (MathSciNet): MR1031787
Zentralblatt MATH: 0693.55002
Digital Object Identifier: doi:10.1007/BFb0086445
S. Husseini, Generalized Lefschetz numbers, Trans. Amer. Math. Soc., 272 (1982), 247–274.
Mathematical Reviews (MathSciNet): MR656489
Zentralblatt MATH: 0507.55001
Digital Object Identifier: doi:10.1090/S0002-9947-1982-0656489-X
B. Jiang, Lectures on Nielsen Fixed Point Theory, Contemp. Math., 14, Amer. Math. Soc., Providence, 1983.
Mathematical Reviews (MathSciNet): MR685755
Zentralblatt MATH: 0512.55003
B. Jiang, A characterization of fixed point classes, Fixed Point Theory and Its Applications, Contemp. Math., 72, Amer. Math. Soc., Providence, 1988, pp. 157–160.
Mathematical Reviews (MathSciNet): MR956486
Zentralblatt MATH: 0648.55003
B. Jiang, A primer of Nielsen fixed point theory, Handbook of Topological Fixed Point Theory (eds. R. F. Brown, M. Furi, L. Górniewicz and B. Jiang), Springer, 2005, pp. 617–645.
Mathematical Reviews (MathSciNet): MR2171118
Zentralblatt MATH: 1081.55003
Digital Object Identifier: doi:10.1007/1-4020-3222-6_16
B. Jiang and J. Guo, Fixed points of surface diffeomorphisms, Pacific J. Math., 160 (1993), 67–89.
Mathematical Reviews (MathSciNet): MR1227504
Zentralblatt MATH: 0829.55001
Project Euclid: euclid.pjm/1102624565
J. Los, Pseudo-Anosov maps and invariant train tracks in the disc: A finite algorithm, Proc. London Math. Soc., 66 (1993), 400–430.
Mathematical Reviews (MathSciNet): MR1199073
Zentralblatt MATH: 0788.58039
Digital Object Identifier: doi:10.1112/plms/s3-66.2.400
T. Matsuoka, The Burau representation of the braid group and the Nielsen-Thurston classification, Nielsen Theory and Dynamical Systems (ed. C. McCord), Contemp. Math., 152, Amer. Math. Soc., Providence, 1993, pp. 229–248.
Mathematical Reviews (MathSciNet): MR1243478
Zentralblatt MATH: 0791.57010
S. Moran, The Mathematical Theory of Knots and Braids: an Introduction, North-Holland Math. Studies, 82, North-Holland, Amsterdam, 1983.
Mathematical Reviews (MathSciNet): MR738974
Zentralblatt MATH: 0528.57001
K. Reidemeister, Automorphismen von Homotopiekettenringen, Math. Ann., 112 (1936), 586–593.
Mathematical Reviews (MathSciNet): MR1513064
Digital Object Identifier: doi:10.1007/BF01565432
W. P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc., 19 (1988), 417–431.
Mathematical Reviews (MathSciNet): MR956596
Digital Object Identifier: doi:10.1090/S0273-0979-1988-15685-6
Project Euclid: euclid.bams/1183554722
J. Wagner, An algorithm for calculating the Nielsen number on surfaces with boundary, Trans. Amer. Math. Soc., 351 (1999), 41–62.
Mathematical Reviews (MathSciNet): MR1401531
Zentralblatt MATH: 0910.55001
Digital Object Identifier: doi:10.1090/S0002-9947-99-01827-9
JSTOR: links.jstor.org
F. Wecken, Fixpunktklassen II, Math. Ann., 118 (1941), 216–234.
Mathematical Reviews (MathSciNet): MR10280
Digital Object Identifier: doi:10.1007/BF01487362
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