Applications of Nielsen theory to dynamics
Banach Center Publications (1999)
- Volume: 49, Issue: 1, page 203-221
- ISSN: 0137-6934
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top- [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
- [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.
- [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
- [BH1] M. Bestvina and M. Handel, Train tracks and automorphisms of free groups, Ann. of Math. 135 (1992) 1-51. Zbl0757.57004
- [BH2] M. Bestvina and M. Handel, Train tracks for surface automorphisms, Topology 34 (1995) 109-140. Zbl0837.57010
- [Bi] J. S. Birman, Braids, Links, and Mapping Class Groups, Ann. Math. Stud. 82, Princeton Univ. Press, Princeton, 1974.
- [Bo1] P. Boyland, Topological methods in surface dynamics, Topology Appl. 58 (1994) 223-298. Zbl0810.54031
- [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.
- [BF] P. Boyland and J. Franks, Notes on Dynamics of Surface Homeomorphisms, informal lecture notes, Univ. of Warwick, Coventry, 1989.
- [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
- [B] R. F. Brown, The Lefschetz Fixed Point Theorem, Scott-Foresman, Chicago, 1971. Zbl0216.19601
- [C] M. Chas, Minimum periods of homeomorphisms on orientable surfaces, thesis, UAB Barcelona, 1998.
- [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
- [D] A. Dold, Lectures on Algebraic Topology, Springer, Berlin, 1972, 1980. Zbl0234.55001
- [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
- [FLP] A. Fathi, F. Laudenbach and V. Poenaru, Travaux de Thurston sur les surfaces, Astérisque 66-67 (1979).
- [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
- [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
- [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.
- [Fu1] F. B. Fuller, The existence of periodic points, Ann. of Math. 57 (1953) 229-230. Zbl0050.17203
- [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.
- [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. Zbl0807.19004
- [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
- [Ha] B. Halpern, Fixed points for iterates, Pacific J. Math. 25 (1968) 255-275. Zbl0157.30201
- [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
- [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. Zbl0515.54016
- [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. Zbl1039.55501
- [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
- [J4] B. Jiang, Bounds for fixed points on surfaces, Math. Ann. 311 (1998), 467-479. Zbl0903.55003
- [JG] B. Jiang and B. Guo, Fixed points of surface diffeomorphisms, Pacific J. Math. 160 (1993) 67-89. Zbl0829.55001
- [JL] B. Jiang and J. Llibre, Minimal sets of periods for torus maps, Discrete Cont. Dynam. Systems 4 (1998), 301-320. Zbl0965.37019
- [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. Zbl0928.57006
- [Ko1] B. Kolev, Entropie topologique et représentation de Burau, C. R. Acad. Sci. Paris 309 (1989) 835-838. Zbl0688.58021
- [Ko2] B. Kolev, Point fixe lié à une orbite périodique d’un difféomorphisme de , C. R. Acad. Sci. Paris 310 (1990) 831-833.
- [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.
- [Li] X.-S. Lin, Representations of knot groups and twisted Alexander polynomials, preprint, 1990.
- [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
- [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.
- [M] S. Moran, The Mathematical Theory of Knots and Braids, An Introduction, North-Holland Math. Stud. 82, North-Holland, Amsterdam, 1983.
- [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
- [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. Zbl0553.55001
- [T] W. P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. 19 (1988) 417-431. Zbl0674.57008
- [Wd] M. Wada, Twisted Alexander polynomial for finitely presentable groups, Topology 33 (1994) 241-256. Zbl0822.57006
- [Wa1] S. Wang, Maximum orders of periodic maps on closed surfaces, Topology Appl. 41 (1991) 255-262.
- [Wa2] S. Wang, Free degrees of homeomorphisms and periodic maps on closed surfaces, Topology Appl. 46 (1992) 81-87.
- [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.