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1999 | 49 | 1 | 203-221

Tytuł artykułu

Applications of Nielsen theory to dynamics

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Języki publikacji

EN

Abstrakty

EN
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.

Słowa kluczowe

Rocznik

Tom

49

Numer

1

Strony

203-221

Opis fizyczny

Daty

wydano
1999

Twórcy

autor
  • Department of Mathematics, Peking University, Beijing 100871, China

Bibliografia

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  • [BH1] M. Bestvina and M. Handel, Train tracks and automorphisms of free groups, Ann. of Math. 135 (1992) 1-51.
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  • [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.
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  • [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.
  • [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.
  • [Gu] J. Guaschi, Representations of Artin's braid groups and linking numbers of periodic orbits, J. Knot Theory Ramif. 4 (1995) 197-212.
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  • [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.
  • [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.
  • [I] N. V. Ivanov, Entropy and the Nielsen numbers, Soviet Math. Dokl. 26 (1982) 63-66.
  • [J1] B. Jiang, Lectures on Nielsen Fixed Point Theory, Contemp. Math. 14, Amer. Math. Soc., Providence, 1983.
  • [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.
  • [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.
  • [J4] B. Jiang, Bounds for fixed points on surfaces, Math. Ann. 311 (1998), 467-479.
  • [JG] B. Jiang and B. Guo, Fixed points of surface diffeomorphisms, Pacific J. Math. 160 (1993) 67-89.
  • [JL] B. Jiang and J. Llibre, Minimal sets of periods for torus maps, Discrete Cont. Dynam. Systems 4 (1998), 301-320.
  • [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.
  • [K] T.-H. Kiang, The Theory of Fixed Point Classes, Science Press, Beijing, 1979, 1986 (in Chinese); English edition, Springer, Berlin, 1989.
  • [KL] P. Kirk and C. Livingston, Twisted knot polynomials: inversion, mutation and concordance, Topology 38 (1999), 663-671.
  • [Ko1] B. Kolev, Entropie topologique et représentation de Burau, C. R. Acad. Sci. Paris 309 (1989) 835-838.
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  • [Li] X.-S. Lin, Representations of knot groups and twisted Alexander polynomials, preprint, 1990.
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  • [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.
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  • [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.
  • [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.
  • [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.
  • [R] K. Reidemeister, Automorphismen von Homotopiekettenringen, Math. Ann. 112 (1936) 586-593.
  • [S] H. Schirmer, A relative Nielsen number, Pacific J. Math. 122 (1986) 459-473.
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  • [Wa1] S. Wang, Maximum orders of periodic maps on closed surfaces, Topology Appl. 41 (1991) 255-262.
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  • [W] F. Wecken, Fixpunktklassen, I, Math. Ann. 117 (1941) 659-671; II, 118 (1942) 216-234; III, 118 (1942) 544-577.
  • [Y] C. Y. You, A note on periodic points on tori, Beijing Math. 1 (1995) 224-230.

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