Applications of Nielsen theory to dynamics

Jiang, Boju

Banach Center Publications, Tome 50 (1999), p. 203-221 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 1999-01-01

Zbl 0959.55002

EUDML-ID : urn:eudml:doc:208961

Bibliographie

[000] [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. | Zbl 0804.54032

[001] [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.

[002] [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. | Zbl 0804.57005

[003] [BH1] M. Bestvina and M. Handel, Train tracks and automorphisms of free groups, Ann. of Math. 135 (1992) 1-51. | Zbl 0757.57004

[004] [BH2] M. Bestvina and M. Handel, Train tracks for surface automorphisms, Topology 34 (1995) 109-140. | Zbl 0837.57010

[005] [Bi] J. S. Birman, Braids, Links, and Mapping Class Groups, Ann. Math. Stud. 82, Princeton Univ. Press, Princeton, 1974.

[006] [Bo1] P. Boyland, Topological methods in surface dynamics, Topology Appl. 58 (1994) 223-298. | Zbl 0810.54031

[007] [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.

[008] [BF] P. Boyland and J. Franks, Notes on Dynamics of Surface Homeomorphisms, informal lecture notes, Univ. of Warwick, Coventry, 1989.

[009] [Br] L. Brouwer, Über die Minimalzahl der Fixpunkte bei den Klassen von eindeutigen stetigen Transformationen der Ringflächen, Math. Ann. 82 (1921) 94-96. | Zbl 47.0528.01

[010] [B] R. F. Brown, The Lefschetz Fixed Point Theorem, Scott-Foresman, Chicago, 1971. | Zbl 0216.19601

[011] [C] M. Chas, Minimum periods of homeomorphisms on orientable surfaces, thesis, UAB Barcelona, 1998.

[012] [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. | Zbl 0853.55001

[013] [D] A. Dold, Lectures on Algebraic Topology, Springer, Berlin, 1972, 1980. | Zbl 0234.55001

[014] [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. | Zbl 0563.55001

[015] [FLP] A. Fathi, F. Laudenbach and V. Poenaru, Travaux de Thurston sur les surfaces, Astérisque 66-67 (1979).

[016] [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. | Zbl 0737.58048

[017] [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. | Zbl 0793.58029

[018] [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.

[019] [Fu1] F. B. Fuller, The existence of periodic points, Ann. of Math. 57 (1953) 229-230. | Zbl 0050.17203

[020] [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.

[021] [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.

[022] [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. | Zbl 0807.19004

[023] [Gu] J. Guaschi, Representations of Artin's braid groups and linking numbers of periodic orbits, J. Knot Theory Ramif. 4 (1995) 197-212. | Zbl 0844.57013

[024] [Ha] B. Halpern, Fixed points for iterates, Pacific J. Math. 25 (1968) 255-275. | Zbl 0157.30201

[025] [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. | Zbl 54.0610.01

[026] [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.

[027] [I] N. V. Ivanov, Entropy and the Nielsen numbers, Soviet Math. Dokl. 26 (1982) 63-66. | Zbl 0515.54016

[028] [J1] B. Jiang, Lectures on Nielsen Fixed Point Theory, Contemp. Math. 14, Amer. Math. Soc., Providence, 1983.

[029] [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. | Zbl 1039.55501

[030] [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. | Zbl 0798.55001

[031] [J4] B. Jiang, Bounds for fixed points on surfaces, Math. Ann. 311 (1998), 467-479. | Zbl 0903.55003

[032] [JG] B. Jiang and B. Guo, Fixed points of surface diffeomorphisms, Pacific J. Math. 160 (1993) 67-89. | Zbl 0829.55001

[033] [JL] B. Jiang and J. Llibre, Minimal sets of periods for torus maps, Discrete Cont. Dynam. Systems 4 (1998), 301-320. | Zbl 0965.37019

[034] [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.

[035] [K] T.-H. Kiang, The Theory of Fixed Point Classes, Science Press, Beijing, 1979, 1986 (in Chinese); English edition, Springer, Berlin, 1989.

[036] [KL] P. Kirk and C. Livingston, Twisted knot polynomials: inversion, mutation and concordance, Topology 38 (1999), 663-671. | Zbl 0928.57006

[037] [Ko1] B. Kolev, Entropie topologique et représentation de Burau, C. R. Acad. Sci. Paris 309 (1989) 835-838. | Zbl 0688.58021

[038] [Ko2] B. Kolev, Point fixe lié à une orbite périodique d’un difféomorphisme de $ℝ^2$, C. R. Acad. Sci. Paris 310 (1990) 831-833.

[039] [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.

[040] [Li] X.-S. Lin, Representations of knot groups and twisted Alexander polynomials, preprint, 1990.

[041] [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. | Zbl 0788.58039

[042] [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.

[043] [M] S. Moran, The Mathematical Theory of Knots and Braids, An Introduction, North-Holland Math. Stud. 82, North-Holland, Amsterdam, 1983.

[044] [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. | Zbl 47.0527.03

[045] [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.

[046] [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.

[047] [R] K. Reidemeister, Automorphismen von Homotopiekettenringen, Math. Ann. 112 (1936) 586-593.

[048] [S] H. Schirmer, A relative Nielsen number, Pacific J. Math. 122 (1986) 459-473. | Zbl 0553.55001

[049] [T] W. P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. 19 (1988) 417-431. | Zbl 0674.57008

[050] [Wd] M. Wada, Twisted Alexander polynomial for finitely presentable groups, Topology 33 (1994) 241-256. | Zbl 0822.57006

[051] [Wa1] S. Wang, Maximum orders of periodic maps on closed surfaces, Topology Appl. 41 (1991) 255-262.

[052] [Wa2] S. Wang, Free degrees of homeomorphisms and periodic maps on closed surfaces, Topology Appl. 46 (1992) 81-87.

[053] [W] F. Wecken, Fixpunktklassen, I, Math. Ann. 117 (1941) 659-671; II, 118 (1942) 216-234; III, 118 (1942) 544-577.

[054] [Y] C. Y. You, A note on periodic points on tori, Beijing Math. 1 (1995) 224-230.