Applications of Nielsen theory to dynamics

Boju Jiang

Banach Center Publications (1999)

  • Volume: 49, Issue: 1, page 203-221
  • ISSN: 0137-6934

Abstract

top
In this talk, we shall look at the application of Nielsen theory to certain questions concerning the "homotopy minimum" or "homotopy stability" of periodic orbits under deformations of the dynamical system. These applications are mainly to the dynamics of surface homeomorphisms, where the geometry and algebra involved are both accessible.

How to cite

top

Jiang, Boju. "Applications of Nielsen theory to dynamics." Banach Center Publications 49.1 (1999): 203-221. <http://eudml.org/doc/208961>.

@article{Jiang1999,
abstract = {In this talk, we shall look at the application of Nielsen theory to certain questions concerning the "homotopy minimum" or "homotopy stability" of periodic orbits under deformations of the dynamical system. These applications are mainly to the dynamics of surface homeomorphisms, where the geometry and algebra involved are both accessible.},
author = {Jiang, Boju},
journal = {Banach Center Publications},
keywords = {Nielsen theory; periodic orbit; dynamical system; homotopy stability},
language = {eng},
number = {1},
pages = {203-221},
title = {Applications of Nielsen theory to dynamics},
url = {http://eudml.org/doc/208961},
volume = {49},
year = {1999},
}

TY - JOUR
AU - Jiang, Boju
TI - Applications of Nielsen theory to dynamics
JO - Banach Center Publications
PY - 1999
VL - 49
IS - 1
SP - 203
EP - 221
AB - In this talk, we shall look at the application of Nielsen theory to certain questions concerning the "homotopy minimum" or "homotopy stability" of periodic orbits under deformations of the dynamical system. These applications are mainly to the dynamics of surface homeomorphisms, where the geometry and algebra involved are both accessible.
LA - eng
KW - Nielsen theory; periodic orbit; dynamical system; homotopy stability
UR - http://eudml.org/doc/208961
ER -

References

top
  1. [ABLSS] L. Alsedà, S. Baldwin, J. Llibre, R. Swanson and W. Szlenk, Torus maps and Nielsen numbers, in: Nielsen Theory and Dynamical Systems, Ch. McCord (ed.), Contemp. Math. 152, Amer. Math. Soc., Providence, 1993, 1-7; Minimal sets of periods for torus maps via Nielsen numbers, Pacific J. Math. 169 (1995), 1-32. Zbl0804.54032
  2. [AF] D. Asimov and J. Franks, Unremovable closed orbits, in: Geometric Dynamics, J. Palis Jr. (ed.), Lecture Notes in Math. 1007, Springer, Berlin, 1983, 22-29. 
  3. [BGN] D. Benardete, M. Gutierrez and Z. Nitecki, A combinatorial approach to reducibility of mapping classes, in: Mapping Class Groups and Moduli Spaces of Riemann Surfaces, C.-F. Bödigheimer, R. M. Hain (eds.), Contemp. Math. 150, Amer. Math. Soc., Providence, 1993, 1-31; Braids and the Nielsen-Thurston classification, J. Knot Theory Ramif. 4 (1995), 549-618. Zbl0804.57005
  4. [BH1] M. Bestvina and M. Handel, Train tracks and automorphisms of free groups, Ann. of Math. 135 (1992) 1-51. Zbl0757.57004
  5. [BH2] M. Bestvina and M. Handel, Train tracks for surface automorphisms, Topology 34 (1995) 109-140. Zbl0837.57010
  6. [Bi] J. S. Birman, Braids, Links, and Mapping Class Groups, Ann. Math. Stud. 82, Princeton Univ. Press, Princeton, 1974. 
  7. [Bo1] P. Boyland, Topological methods in surface dynamics, Topology Appl. 58 (1994) 223-298. Zbl0810.54031
  8. [Bo2] P. Boyland, Isotopy stability of dynamics on surfaces, preprint, in: Geometry and Topology in Dynamics, M. Barge and K. Kuperberg (eds.), Contemp. Math. 246, Amer. Math. Soc., 1999, 17-46. 
  9. [BF] P. Boyland and J. Franks, Notes on Dynamics of Surface Homeomorphisms, informal lecture notes, Univ. of Warwick, Coventry, 1989. 
  10. [Br] L. Brouwer, Über die Minimalzahl der Fixpunkte bei den Klassen von eindeutigen stetigen Transformationen der Ringflächen, Math. Ann. 82 (1921) 94-96. Zbl47.0528.01
  11. [B] R. F. Brown, The Lefschetz Fixed Point Theorem, Scott-Foresman, Chicago, 1971. Zbl0216.19601
  12. [C] M. Chas, Minimum periods of homeomorphisms on orientable surfaces, thesis, UAB Barcelona, 1998. 
  13. [DL] W. Dicks and J. Llibre, Orientation-preserving self-homeomorphisms of the surface of genus two have points of period at most two, Proc. Amer. Math. Soc. 124 (1996), 1583-1591. Zbl0853.55001
  14. [D] A. Dold, Lectures on Algebraic Topology, Springer, Berlin, 1972, 1980. Zbl0234.55001
  15. [FH] E. Fadell and S. Husseini, The Nielsen number on surfaces, in: Topological Methods in Nonlinear Functional Analysis, S. P. Singh et al. (eds.), Contemp. Math. 21, Amer. Math. Soc., Providence, 1983, 59-98. Zbl0563.55001
  16. [FLP] A. Fathi, F. Laudenbach and V. Poenaru, Travaux de Thurston sur les surfaces, Astérisque 66-67 (1979). 
  17. [FL] J. Franks and J. Llibre, Periods of surface homeomorphisms, in: Continuum Theory and Dynamical Systems, M. Brown (ed.), Contemp. Math. 117, Amer. Math. Soc., Providence, 1991, 63-77. Zbl0737.58048
  18. [FM] J. Franks and M. Misiurewicz, Cycles for disk homeomorphisms and thick trees, in: Nielsen Theory and Dynamical Systems, Ch. McCord (ed.), Contemp. Math. 152, Amer. Math. Soc., Providence, 1993, 69-139. Zbl0793.58029
  19. [Fr] D. Fried, Periodic points and twisted coefficients, in: Geometric Dynamics, J. Palis Jr. (ed.), Lecture Notes in Math. 1007, Springer, Berlin, 1983, 261-293; Homological identities for closed orbits, Invent. Math. 71 (1983) 419-442; Entropy and twisted cohomology, Topology 25 (1986) 455-470; Lefschetz formulas for flows, in: The Lefschetz Centennial Conference, Part III: Proceedings on Differential Equations, A. Verjovsky (ed.), Contemp. Math. 58.III, Amer. Math. Soc., Providence, 1987, 19-69. 
  20. [Fu1] F. B. Fuller, The existence of periodic points, Ann. of Math. 57 (1953) 229-230. Zbl0050.17203
  21. [Fu2] F. B. Fuller, The treatment of periodic orbits by the methods of fixed point theory, Bull. Amer. Math. Soc. 72 (1966) 838-840; An index of fixed point type for periodic orbits, Amer. J. Math. 89 (1967) 133-148. 
  22. [GST] J.-M. Gambaudo, S. van Strien and C. Tresser, Vers un ordre de Sarkovskii pour les plongements du disque préservant l'orientation, C. R. Acad. Sci. Paris Sér. I 310 (1990) 291-294. 
  23. [GN] R. Geoghegan and A. Nicas, Lefschetz trace formulae, zeta functions and torsion in dynamics, in: Nielsen Theory and Dynamical Systems, Ch. McCord (ed.), Contemp. Math. 152, Amer. Math. Soc., Providence, 1993, 141-157; Trace and torsion in the theory of flows, Topology 33 (1994) 683-719. Zbl0807.19004
  24. [Gu] J. Guaschi, Representations of Artin's braid groups and linking numbers of periodic orbits, J. Knot Theory Ramif. 4 (1995) 197-212. Zbl0844.57013
  25. [Ha] B. Halpern, Fixed points for iterates, Pacific J. Math. 25 (1968) 255-275. Zbl0157.30201
  26. [H] H. Hopf, A new proof of the Lefschetz formula on invariant points, Proc. Nat. Acad. Sci. USA 14 (1928) 149-153; Über die algebraische Anzahl von Fixpunkten, Math. Z. 29 (1929) 493-524. Zbl54.0610.01
  27. [HJ] H.-H. Huang and B.-J. Jiang, Braids and periodic solutions, in: Topological Fixed Point Theory and Applications (Tianjin, 1988), B. Jiang (ed.), Lecture Notes in Math. 1411, Springer, Berlin, 1989, 107-123. 
  28. [I] N. V. Ivanov, Entropy and the Nielsen numbers, Soviet Math. Dokl. 26 (1982) 63-66. Zbl0515.54016
  29. [J1] B. Jiang, Lectures on Nielsen Fixed Point Theory, Contemp. Math. 14, Amer. Math. Soc., Providence, 1983. 
  30. [J2] B. Jiang, Periodic orbits on surfaces via Nielsen fixed point theory, in: Topology-Hawaii, K. H. Dovermann (ed.), World Scientific, Singapore, 1992, 101-118. Zbl1039.55501
  31. [J3] B. Jiang, Nielsen theory for periodic orbits and applications to dynamical systems, in: Nielsen Theory and Dynamical Systems, Ch. McCord (ed.), Contemp. Math. 152, Amer. Math. Soc., Providence, 1993, 183-202; Estimation of the number of periodic orbits, Pacific J. Math. 172 (1996) 151-185. Zbl0798.55001
  32. [J4] B. Jiang, Bounds for fixed points on surfaces, Math. Ann. 311 (1998), 467-479. Zbl0903.55003
  33. [JG] B. Jiang and B. Guo, Fixed points of surface diffeomorphisms, Pacific J. Math. 160 (1993) 67-89. Zbl0829.55001
  34. [JL] B. Jiang and J. Llibre, Minimal sets of periods for torus maps, Discrete Cont. Dynam. Systems 4 (1998), 301-320. Zbl0965.37019
  35. [JW] B. Jiang and S. Wang, Twisted topological invariants associated with representations, in: Topics in Knot Theory, M. E. Bozhüyük (ed.) Kluwer, Dordrecht, 1993, 211-227. 
  36. [K] T.-H. Kiang, The Theory of Fixed Point Classes, Science Press, Beijing, 1979, 1986 (in Chinese); English edition, Springer, Berlin, 1989. 
  37. [KL] P. Kirk and C. Livingston, Twisted knot polynomials: inversion, mutation and concordance, Topology 38 (1999), 663-671. Zbl0928.57006
  38. [Ko1] B. Kolev, Entropie topologique et représentation de Burau, C. R. Acad. Sci. Paris 309 (1989) 835-838. Zbl0688.58021
  39. [Ko2] B. Kolev, Point fixe lié à une orbite périodique d’un difféomorphisme de 2 , C. R. Acad. Sci. Paris 310 (1990) 831-833. 
  40. [L] S. Lefschetz, Continuous transformations of manifolds, Proc. Nat. Acad. Sci. USA 9 (1923) 90-93; 11 (1925) 290-292; Intersections and transformations of complexes and manifolds, Trans. Amer. Math. Soc. 28 (1926) 1-49. 
  41. [Li] X.-S. Lin, Representations of knot groups and twisted Alexander polynomials, preprint, 1990. 
  42. [Lo] J. E. Los, Pseudo-Anosov maps and invariant train tracks in the disk: a finite algorithm, Proc. London Math. Soc. 66 (1993) 400-430. Zbl0788.58039
  43. [Ma ] T. Matsuoka, The number and linking of periodic solutions of periodic systems, Invent. Math. 70 (1983) 319-340; Waveform in dynamical systems of ordinary differential equations, Japan. J. Appl. Math. 1 (1984) 417-434. 
  44. [M] S. Moran, The Mathematical Theory of Knots and Braids, An Introduction, North-Holland Math. Stud. 82, North-Holland, Amsterdam, 1983. 
  45. [N1] J. Nielsen, Über die Minimalzahl der Fixpunkte bei den Abbildungstypen der Ringflächen, Math. Ann. 82 (1921) 83-93; also in: Jakob Nielsen: Collected Mathematical Papers, vol. 1, V. L. Hansen (ed.), Birkhäuser, Boston, 1986, 99-109. Zbl47.0527.03
  46. [N2] J. Nielsen, Untersuchungen zur Topologie des geschlossenen zweiseitigen Flächen, I, Acta Math. 50 (1927) 189-358; English transl.: Investigations in the topology of closed orientable surfaces, I, in: Jakob Nielsen: Collected Mathematical Papers, vol. 1, V. L. Hansen (ed.), Birkhäuser, Boston, 1986, 223-341. 
  47. [N3] J. Nielsen, Fixpunktfrie afbildninger, Mat. Tidsskr. B (1942) 25-41, reviewed by R. Fox, Math. Reviews 7 (1946), 137; English transl.: Fixed point free mappings, in: Jakob Nielsen: Collected Mathematical Papers, vol. 2, V. L. Hansen (ed.), Birkhäuser, Boston, 1986, 221-232. 
  48. [R] K. Reidemeister, Automorphismen von Homotopiekettenringen, Math. Ann. 112 (1936) 586-593. 
  49. [S] H. Schirmer, A relative Nielsen number, Pacific J. Math. 122 (1986) 459-473. Zbl0553.55001
  50. [T] W. P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. 19 (1988) 417-431. Zbl0674.57008
  51. [Wd] M. Wada, Twisted Alexander polynomial for finitely presentable groups, Topology 33 (1994) 241-256. Zbl0822.57006
  52. [Wa1] S. Wang, Maximum orders of periodic maps on closed surfaces, Topology Appl. 41 (1991) 255-262. 
  53. [Wa2] S. Wang, Free degrees of homeomorphisms and periodic maps on closed surfaces, Topology Appl. 46 (1992) 81-87. 
  54. [W] F. Wecken, Fixpunktklassen, I, Math. Ann. 117 (1941) 659-671; II, 118 (1942) 216-234; III, 118 (1942) 544-577. 
  55. [Y] C. Y. You, A note on periodic points on tori, Beijing Math. 1 (1995) 224-230. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.