Skip to main content

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. L. Alsedà, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One,, 2nd ed., Adv. Ser. Nonlinear Dynam., vol. 5, World Scientific, Singapore, 2000.

    Google Scholar 

  2. J. Andres, A nontrivial example of application of the Nielsen fixed-point theory to differential systems: Problem of Jean Leray, Proc. Amer. Math. Soc. 128 (2000), 2921–2931.

    Article  MATH  MathSciNet  Google Scholar 

  3. _____, Nielsen number, Artin braids, Poincaré operators and multiple nonlinear oscillations, Nonlinear Anal. 47 (2001), 1017–1028.

    Article  MATH  MathSciNet  Google Scholar 

  4. J. Andres and L. Górniewicz, Topological Fixed Point Principles for Boundary Value Problems, “Topological Fixed Point Theory and its Applications”, vol. 1, Kluwer, Dordrecht, Boston, London, 2003.

    Google Scholar 

  5. D. Asimov and J. Franks, Unremovable closed orbits, preprint (1989), (This is a revised version of a paper which appeared in Lecture Notes in Math. Vol. 1007, Springer-Verlag, 1983).

    Google Scholar 

  6. D. Benardete, M. Gutierrez and Z. Nitecki, A combinatorial approach to reducibility of mapping classes, Mapping Class Groups and Moduli Spaces of Riemann Surfaces (C.-F. Bödigheimer and R. Hain, eds.); Contemp. Math. 150 (1993), 1–31.

    Google Scholar 

  7. _____, Braids and the Nielsen-Thurston classification, J. Knot Theory Ramifications 4 (1995), 549–618.

    Article  MathSciNet  Google Scholar 

  8. M. Bestvina and M. Handel, Train-tracks for surface homeomorphisms, Topology 34 (1995), 109–140.

    Article  MathSciNet  MATH  Google Scholar 

  9. J. S. Birman, Braids, links and mapping class groups, Ann. Math. Stud., vol. 82, Princeton Univ. Press, Princeton, 1974.

    Google Scholar 

  10. C. Bonatti and B. Kolev, Existence de points fixes enlacés à une orbite périodique d'un homéomorphisme du plan, Ergodic Theory Dynam. Systems 12 (1992), 677–682.

    MathSciNet  MATH  Google Scholar 

  11. R. Bowen, Entropy and the fundamental group, The Structure of Attractors in Dynamical Systems, Lecture Notes in Math. (N. G. Markley et al., eds.), vol. 668, Springer-Verlag, Berlin—Heidelberg—New York, 1978, pp. 21–29.

    Google Scholar 

  12. P. Boyland, Braid types and a topological method of proving positive entropy (1984), unpublished.

    Google Scholar 

  13. _____, An analog of Sharkovski's theorem for twist maps, Contemp. Math. 81 (1988), 119–133.

    Google Scholar 

  14. _____, Rotation sets and monotone periodic orbits for annulus homeomorphisms, Comm. Math. Helv. 67 (1992), 203–213.

    Google Scholar 

  15. _____, Topological methods in surface dynamics, Topology Appl. 58 (1994), 223–298.

    Google Scholar 

  16. _____, Isotopy stability of dynamics on surfaces, Contemp. Math. 246 (1999), 17–45.

    Google Scholar 

  17. _____, Dynamics of two-dimensional time-periodic Euler fluid flows, preprint.

    Google Scholar 

  18. P. Boyland, H. Aref and M. Stremler, Topological fluid mechanics of stirring, J. Fluid Mech. 403 (2000), 277–304.

    Article  MathSciNet  MATH  Google Scholar 

  19. P. Boyland and J. Franks, Notes on Dynamics of Surface Homeomorphisms, Informal Lectures Notes, Math. Institute, University of Warwick, 1989.

    Google Scholar 

  20. P. Boyland, J. Guaschi and T. Hall, L'ensemble de rotation des homéomorphismes pseudo-Anosov, C. R. Acad. Sci. Paris, Sér. I 316 (1993), 1077–1080.

    MathSciNet  MATH  Google Scholar 

  21. P. Boyland and T. Hall, Isotopy stable dynamics relative to compact invariant sets, Proc. London Math. Soc. 79 (1999), 673–693.

    Article  MathSciNet  MATH  Google Scholar 

  22. P. Boyland, M. Stremler and H. Aref, Topological fluid mechanics of point vortex motions, Phys. D 175 (2003), 69–95.

    Article  MathSciNet  MATH  Google Scholar 

  23. R. F. Brown, The Lefschetz Fixed Point Theorem, Scott—Foresman, Glenview, 1971.

    MATH  Google Scholar 

  24. A. Casson and S. Bleiler, Automorphisms of Surfaces After Nielsen and Thurston, London Math. Soc. Student Texts, No. 9, Cambridge Univ. Press, Cambridge, 1988.

    Google Scholar 

  25. E. N. Dancer and R. Ortega, The index of Lyapunov stable fixed points in two dimensions, J. Dynam. Differential Equations 6 (1994), 631–637.

    Article  MathSciNet  MATH  Google Scholar 

  26. A. de Carvalho and T. Hall, The forcing relation for horseshoe braid types, Experiment. Math. 11 (2002), 271–288.

    MathSciNet  MATH  Google Scholar 

  27. _____, Conjugacies between horseshoe braids, Nonlinearity 16 (2003), 1329–1338.

    Article  MathSciNet  MATH  Google Scholar 

  28. _____, Braid forcing and star-shaped train tracks, Topology 43 (2004), 247–287.

    Article  MathSciNet  MATH  Google Scholar 

  29. E. Fadell and S. Husseini, Fixed point theory for non-simply connected manifolds, Topology 20 (1981), 53–92.

    Article  MathSciNet  MATH  Google Scholar 

  30. _____, The Nielsen number on surfaces, Topological Methods in Nonlinear Functional Analysis (S. P. Singh et al., eds.), Contemp. Math, vol. 21, Amer. Math. Soc., Providence, 1983, pp. 59–98.

    Google Scholar 

  31. A. Fathi, F. Laudenbach and V. Poénaru, Travaux de Thurston sur les surfaces, Astérisque, vol. 66–67, Soc. Math. France, Paris, 1979.

    Google Scholar 

  32. J. Fehrenbach and J. Los, Une minoration de l'entropie topologique des difféomorphismes du disque, J. London Math. Soc. 60 (1999), 912–924.

    Article  MathSciNet  MATH  Google Scholar 

  33. J. Franks, Knots, links and symbolic dynamics, Ann. of Math. 113 (1981), 529–552.

    Article  MATH  MathSciNet  Google Scholar 

  34. _____, Recurrence and fixed points of surface homeomorphisms, Ergodic Theory Dynam. Systems 8* (1988), 99–107.

    Article  MATH  MathSciNet  Google Scholar 

  35. _____, Realizing rotation vectors for torus homeomorphisms, Trans. Amer. Math. Soc. 311 (1989), 107–115.

    Article  MATH  MathSciNet  Google Scholar 

  36. _____, Geodesics on S 2 and periodic points of annulus homeomorphisms, Invent. Math. 108 (1992), 403–418.

    Article  MATH  MathSciNet  Google Scholar 

  37. J. Franks and M. Handel, Entropy and exponential growth of π 1 in dimension two, Proc. Amer. Math. Soc. 102 (1988), 753–760.

    Article  MathSciNet  MATH  Google Scholar 

  38. J. Franks and M. Misiurewicz, Cycles for disk homeomorphisms and thick trees, Nielsen Theory and Dynamical Systems (C. McCord, ed.), Contemp. Math., vol. 152, Amer. Math. Soc., Providence, 1993, pp. 69–139.

    Google Scholar 

  39. D. Fried, Periodic points and twisted coefficients, Geometric Dynamics (J. Palis ed.), Lecture Notes in Math., vol. 1007, Springer-Verlag, Berlin—Heidelberg—New York, 1983, pp. 261–293.

    Google Scholar 

  40. _____, Entropy and twisted cohomology, Topology 25 (1986), 455–470.

    Article  MATH  MathSciNet  Google Scholar 

  41. J. M. Gambaudo, Periodic orbits and fixed points of a C 1 orientation-preserving embedding of D 2, Math. Proc. Cambridge Philos. Soc. 108 (1990), 307–310.

    Article  MATH  MathSciNet  Google Scholar 

  42. J. M. Gambaudo, J. Guaschi and T. Hall, Period-multiplying cascades for diffeomorphisms of the disc, Math. Proc. Cambridge Philos. Soc. 116 (1994), 359–374.

    MathSciNet  MATH  Google Scholar 

  43. J. M. Gambaudo and J. Llibre, A note on the periods of surface homeomorphisms, J. Math. Anal. Appl. 177 (1993), 627–632.

    Article  MathSciNet  MATH  Google Scholar 

  44. J. M. Gambaudo, S. van Strien and C. Tresser, The periodic orbit structure of orientation preserving diffeomorphisms on D 2 with topological entropy zero, Ann. Inst. H. Poincaré Anal. Non Linéaire 49 (1989), 335–356.

    Google Scholar 

  45. _____, Vers un ordre de Sarkovskii pour les plongements du disque préservant l'orientation, C. R. Acad. Sci. Paris, Sér. I 310 (1990), 291–294.

    MATH  Google Scholar 

  46. J. Guaschi, Lefschetz numbers of periodic orbits of pseudo-Anosov homeomorphisms, Math. Proc. Cambridge Philos. Soc. 115 (1994), 121–132.

    MATH  MathSciNet  Google Scholar 

  47. _____, Pseudo-Anosov braid types of the disc or sphere of low cardinality imply all periods, J. London Math. Soc. 50 (1994), 594–608.

    MATH  MathSciNet  Google Scholar 

  48. _____, Representations of Artin's braid groups and linking numbers of periodic orbits, J. Knot Theory Ramifications 4 (1995), 197–212.

    Article  MATH  MathSciNet  Google Scholar 

  49. _____, Nielsen theory, braids and fixed points of surface homeomorphisms, Topology Appl. 117 (2002), 199–230.

    Article  MATH  MathSciNet  Google Scholar 

  50. J. Guaschi, J. Llibre and R. S. MacKay, A classification of braid types for periodic orbits of diffeomorphisms of surfaces of genus one with topological entropy zero, Publ. Mat. 35 (1991), 543–558.

    MathSciNet  MATH  Google Scholar 

  51. G. R. Hall, A topological version of a theorem of Mather on twist maps, Ergodic Theory Dynam. Systems 4 (1984), 585–603.

    MATH  MathSciNet  Google Scholar 

  52. T. Hall, Unremovable periodic orbits of homeomorphisms, Math. Proc. Cambridge Philos. Soc. 110 (1991), 523–531.

    MATH  MathSciNet  Google Scholar 

  53. _____, Weak universality in two-dimensional transitions to chaos, Phys. Rev. Lett. 71 (1993), 58–61.

    Article  Google Scholar 

  54. _____, Fat one-dimensional representatives of pseudo-Anosov isotopy classes with minimal periodic orbit structure, Nonlinearity 7 (1994), 367–384.

    Article  MATH  MathSciNet  Google Scholar 

  55. _____, The creation of horseshoes, Nonlinearity 7 (1994), 861–924.

    Article  MATH  MathSciNet  Google Scholar 

  56. M. Handel, The rotation set of a homeomorphism of the annulus is closed, Comm. Math. Phys. 127 (1990), 339–349.

    Article  MATH  MathSciNet  Google Scholar 

  57. _____, The forcing partial order on the three times punctured disk, Ergodic Theory Dynam. Systems 17 (1997), 593–610.

    Article  MATH  MathSciNet  Google Scholar 

  58. V. L. Hansen, Braids and Coverings: Selected Topics, London Math. Soc. Student Texts, vol. 18, Cambridge Univ. Press, Cambridge, 1989.

    Google Scholar 

  59. E. Hayakawa, A sufficient condition for the existence of periodic points of homeomorphisms on surfaces, Tokyo J. Math. 18 (1995), 213–219; Erratum: Tokyo J. Math. 20 (1997), 519.

    MATH  MathSciNet  Google Scholar 

  60. _____, Markov maps on trees, Math. Japonica 51 (2000), 235–240.

    MATH  MathSciNet  Google Scholar 

  61. H. H. Huang and B. J. Jiang, Braids and periodic solutions, Topological Fixed Point Theory and Applications (B. J. Jiang, ed.), Lecture Notes in Math., vol. 1411, Springer—Verlag, Berlin—Heidelberg—New York, 1989, pp. 107–123.

    Google Scholar 

  62. S. Husseini, Generalized Lefschetz numbers, Trans. Amer. Math. Soc. 272 (1982), 247–274.

    Article  MATH  MathSciNet  Google Scholar 

  63. B. Jiang, Lectures on Nielsen Fixed Point Theory, Contemp. Math., vol. 14, Amer. Math. Soc., Providence, 1983.

    Google Scholar 

  64. _____, A characterization of Nielsen classes, Fixed Point Theory and Its Applications, Contemp. Math., vol. 72, Amer. Math. Soc, Providence, 1988, pp. 157–160.

    Google Scholar 

  65. _____, Periodic orbits on surfaces via Nielsen fixed point theory, Topology-Hawaii (K. H. Dovermann, ed.), World Scientific, Singapore, 1991, pp. 101–118.

    Google Scholar 

  66. _____, Nielsen theory for periodic orbits and applications to dynamical systems, Nielsen Theory and Dynamical Systems (C. McCord, ed., ed.), Contemp. Math., vol. 152, Amer. Math. Soc., Providence, 1993, pp. 183–202.

    Google Scholar 

  67. _____, Estimation of the number of periodic orbits, Pacific J. Math. 172 (1996), 151–185.

    MATH  MathSciNet  Google Scholar 

  68. _____, Bounds for fixed points on surfaces, Math. Ann. 311 (1998), 467–479.

    Article  MATH  MathSciNet  Google Scholar 

  69. B. Jiang and J. Guo, Fixed points of surface diffeomorphisms, Pacific J. Math. 160 (1993), 67–89.

    MathSciNet  MATH  Google Scholar 

  70. M. R. Kelly, Computing Nielsen numbers of surface homeomorphisms, Topology 35 (1996), 13–25.

    Article  MATH  MathSciNet  Google Scholar 

  71. E. Kin, The forcing relation on periodic orbits of pseudo-Anosov braid types for disk automorphisms, preprint.

    Google Scholar 

  72. B. Kolev, Entropie topologique et représentation de Burau, C. R. Acad. Sci. Paris Sér. I 309 (1989), 835–838.

    MATH  MathSciNet  Google Scholar 

  73. _____, Point fixe lié à une orbite périodique d'un difféomorphisme de \(\mathbb{R}^2 \) , C. R. Acad. Sci. Paris, Sér. 310 (1990), 831–833.

    MATH  MathSciNet  Google Scholar 

  74. _____, Periodic orbits of period 3 in the disc, Nonlinearity 7 (1994), 1067–1071.

    Article  MATH  MathSciNet  Google Scholar 

  75. I. Kra, On the Nielsen-Thurston-Bers type of some self-maps of Riemann surfaces, Acta Math. 146 (1981), 231–270.

    Article  MATH  MathSciNet  Google Scholar 

  76. J. Kwapisz and R. Swanson, Asymptotic entropy, periodic orbits, and pseudo-Anosov maps, Ergodic Theory Dynam. Systems 18 (1998), 425–439.

    Article  MathSciNet  MATH  Google Scholar 

  77. J. Llibre and R. S. MacKay, A classification of braid types for diffeomorphisms of surfaces of genus zero with topological entropy zero, J. London Math. Soc. 42 (1990), 562–576.

    MathSciNet  MATH  Google Scholar 

  78. _____, Rotation vectors and entropy for homeomorphisms of the torus isotopic to the identity, Ergodic Theory Dynam. Systems 11 (1991), 115–128.

    MathSciNet  Google Scholar 

  79. _____, Pseudo-Anosov homeomorphisms on a sphere with four punctures have all periods, Math. Proc. Cambridge Philos. Soc. 112 (1992), 539–549.

    Article  MathSciNet  MATH  Google Scholar 

  80. J. Los, Pseudo-Anosov maps and invariant train tracks in the disc: A finite algorithm, Proc. London Math. Soc. 66 (1993), 400–430.

    MATH  MathSciNet  Google Scholar 

  81. _____, On the forcing relation for surface homeomorphisms, Publ. Math. IHES 85 (1997), 5–61.

    MATH  MathSciNet  Google Scholar 

  82. S. Matsumoto, Rotation sets of surface homeomorphisms, Bol. Soc. Brasil. Mat. 28 (1997), 89–101.

    Article  MATH  MathSciNet  Google Scholar 

  83. _____, Arnold conjecture for surface homeomorphisms, Topology Appl. 104 (2000), 191–214.

    Article  MATH  MathSciNet  Google Scholar 

  84. T. Matsuoka, The number and linking of periodic solutions of periodic systems, Invent. Math. 70 (1983), 319–340.

    Article  MATH  MathSciNet  Google Scholar 

  85. _____, Waveform in the dynamical study of ordinary differential equations, Japan J. Appl. Math. 1 (1984), 417–434.

    Article  MATH  MathSciNet  Google Scholar 

  86. _____, Braids of periodic points and a 2-dimensional analogue of Sharkovskiĭ's ordering, Dynamical Systems and Nonlinear Oscillations (G. Ikegami, ed.), World Sci. Adv. Ser. Dynam. Systems, vol. 1, World Scientific, Singapore, 1986, pp. 58–72.

    Google Scholar 

  87. _____, The number and linking of periodic solutions of non-dissipative systems, J. Differential Equations 76 (1988), 190–201.

    Article  MATH  MathSciNet  Google Scholar 

  88. _____, The number of periodic points of smooth maps, Ergodic Theory Dynam. Systems 9 (1989), 153–163.

    MATH  MathSciNet  Google Scholar 

  89. _____, The Burau representation of the braid group and the Nielsen-Thurston classification, Nielsen Theory and Dynamical Systems (C. McCord, ed.), Contemp Math., vol. 152, Amer. Math. Soc., Providence, 1993, pp. 229–248.

    Google Scholar 

  90. _____, Braid type and torsion number for fixed points of orientation-preserving embeddings on the disk, Math. Japonica 42 (1995), 25–34.

    MATH  MathSciNet  Google Scholar 

  91. _____, Braid type of the fixed point set for orientation-preserving embeddings on the disk, Tokyo J. Math. 18 (1995), 457–472.

    Article  MATH  MathSciNet  Google Scholar 

  92. _____, Periodic points of disk homeomorphisms having a pseudo-Anosov component, Hokkaido Math. J. 27 (1998), 423–455.

    MATH  MathSciNet  Google Scholar 

  93. _____, Braid invariants and instability of periodic solutions of time-periodic 2-dimensional ODE's, Topol. Methods Nonlinear Anal. 14 (1999), 261–274.

    MATH  MathSciNet  Google Scholar 

  94. _____, On the linking structure of periodic orbits for embeddings of the disk, Math. Japonica 51 (2000), 241–254.

    MATH  MathSciNet  Google Scholar 

  95. _____, Fixed point index and braid invariant for fixed points of embeddings on the disk, Topology Appl. 122 (2002), 337–352.

    Article  MATH  MathSciNet  Google Scholar 

  96. F. A. McRobie and J. M. T. Thompson, Braids and knots in driven oscillators, Internat. J. Bifur. Chaos 3 (1993), 1343–1361.

    Article  MathSciNet  MATH  Google Scholar 

  97. S. B. Miled and J. M. Gambaudo, Cascades of periodic orbits in two dimensions, Nonlinearity 10 (1997), 1627–1641.

    Article  MathSciNet  MATH  Google Scholar 

  98. M. Misiurewicz and K. Ziemian, Rotation sets for maps of tori, J. London Math. Soc. 40 (1989), 490–506.

    MathSciNet  Google Scholar 

  99. _____, Rotation sets and ergodic measures for torus homeomorphisms, Fund. Math. 137 (1991), 45–52.

    MathSciNet  MATH  Google Scholar 

  100. S. Moran, The Mathematical Theory of Knots and Braids: an Introduction, North-Holland Math. Studies, vol. 82, North-Holland, Amsterdam, 1983.

    Google Scholar 

  101. K. Murasugi, On closed 3-braids, Mem. Amer. Math. Soc. 151 (1974).

    Google Scholar 

  102. B. Peckham, The necessity of the Hopf bifurcation for periodically forced oscillators, Nonlinearity 3 (1990), 261–280.

    Article  MATH  MathSciNet  Google Scholar 

  103. M. Pollicott, Rotation sets for homeomorphisms and homology, Trans. Amer. Math. Soc. 331 (1992), 881–894.

    Article  MATH  MathSciNet  Google Scholar 

  104. M. Pollicott and R. Sharp, Growth of periodic points and rotation vectors on surfaces, Topology 36 (1997), 765–774.

    Article  MathSciNet  MATH  Google Scholar 

  105. D. Rolfsen, New developments in the theory of Artin's braid groups, Topology Appl. 127 (2003), 77–90.

    Article  MATH  MathSciNet  Google Scholar 

  106. A. N. Sharkovski\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}\to {i} \) , Coexistence of cycles of a continuous map of a line into itself, Ukrain. Math. Zh. 16 (1964), 61–71. (Russian)

    Google Scholar 

  107. R. Sharp, Periodic points and rotation vectors for torus diffeomorphisms, Topology 34 (1995), 351–357.

    Article  MATH  MathSciNet  Google Scholar 

  108. H. Shiraki, On braid type of fixed points of homeomorphisms defined on the torus, Mem. Fac. Sci. Kôchi Univ. Ser. A Math. 20 (1999), 113–122.

    MATH  MathSciNet  Google Scholar 

  109. H. Shiraki, Fixed point indices of homeomorphisms defined on the torus, Hokkaido Math. J. 32 (2003), 59–74.

    MATH  MathSciNet  Google Scholar 

  110. W. T. Song, K. H. Ko and J. Los, Entropies of braids, J. Knot Theory Ramifications 11 (2002), 647–666.

    Article  MathSciNet  MATH  Google Scholar 

  111. W. P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. 19 (1988), 417–431.

    Article  MATH  MathSciNet  Google Scholar 

  112. J. A. Walsh, Rotation vectors for toral maps and flows: a tutorial, Internat. J. Bifur. Chaos 5 (1995), 321–348.

    Article  MATH  Google Scholar 

  113. Y. Yamaguchi and K. Tanikawa, Symmetrical non-Birkhoff period-3 orbits in standard-like mappings, Progr. Theoret. Phys. 104 (2000), 943–954.

    Article  MathSciNet  Google Scholar 

  114. _____, Non-symmetric non-Birkhoff period-2 orbits in the standard mapping, Progr. Theoret. Phys. 106 (2001), 691–696.

    Article  MathSciNet  MATH  Google Scholar 

  115. _____, Dynamical ordering of non-Birkhoff orbits and topological entropy in the standard mapping, Progr. Theoret. Phys. 107 (2002), 1117–1145.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Springer

About this chapter

Cite this chapter

Matsuoka, T. (2005). Periodic Points and Braid Theory. In: Brown, R.F., Furi, M., Górniewicz, L., Jiang, B. (eds) Handbook of Topological Fixed Point Theory. Springer, Dordrecht. https://doi.org/10.1007/1-4020-3222-6_5

Download citation

Publish with us

Policies and ethics