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References
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.
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.
_____, Nielsen number, Artin braids, Poincaré operators and multiple nonlinear oscillations, Nonlinear Anal. 47 (2001), 1017–1028.
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.
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).
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.
_____, Braids and the Nielsen-Thurston classification, J. Knot Theory Ramifications 4 (1995), 549–618.
M. Bestvina and M. Handel, Train-tracks for surface homeomorphisms, Topology 34 (1995), 109–140.
J. S. Birman, Braids, links and mapping class groups, Ann. Math. Stud., vol. 82, Princeton Univ. Press, Princeton, 1974.
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.
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.
P. Boyland, Braid types and a topological method of proving positive entropy (1984), unpublished.
_____, An analog of Sharkovski's theorem for twist maps, Contemp. Math. 81 (1988), 119–133.
_____, Rotation sets and monotone periodic orbits for annulus homeomorphisms, Comm. Math. Helv. 67 (1992), 203–213.
_____, Topological methods in surface dynamics, Topology Appl. 58 (1994), 223–298.
_____, Isotopy stability of dynamics on surfaces, Contemp. Math. 246 (1999), 17–45.
_____, Dynamics of two-dimensional time-periodic Euler fluid flows, preprint.
P. Boyland, H. Aref and M. Stremler, Topological fluid mechanics of stirring, J. Fluid Mech. 403 (2000), 277–304.
P. Boyland and J. Franks, Notes on Dynamics of Surface Homeomorphisms, Informal Lectures Notes, Math. Institute, University of Warwick, 1989.
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.
P. Boyland and T. Hall, Isotopy stable dynamics relative to compact invariant sets, Proc. London Math. Soc. 79 (1999), 673–693.
P. Boyland, M. Stremler and H. Aref, Topological fluid mechanics of point vortex motions, Phys. D 175 (2003), 69–95.
R. F. Brown, The Lefschetz Fixed Point Theorem, Scott—Foresman, Glenview, 1971.
A. Casson and S. Bleiler, Automorphisms of Surfaces After Nielsen and Thurston, London Math. Soc. Student Texts, No. 9, Cambridge Univ. Press, Cambridge, 1988.
E. N. Dancer and R. Ortega, The index of Lyapunov stable fixed points in two dimensions, J. Dynam. Differential Equations 6 (1994), 631–637.
A. de Carvalho and T. Hall, The forcing relation for horseshoe braid types, Experiment. Math. 11 (2002), 271–288.
_____, Conjugacies between horseshoe braids, Nonlinearity 16 (2003), 1329–1338.
_____, Braid forcing and star-shaped train tracks, Topology 43 (2004), 247–287.
E. Fadell and S. Husseini, Fixed point theory for non-simply connected manifolds, Topology 20 (1981), 53–92.
_____, 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.
A. Fathi, F. Laudenbach and V. Poénaru, Travaux de Thurston sur les surfaces, Astérisque, vol. 66–67, Soc. Math. France, Paris, 1979.
J. Fehrenbach and J. Los, Une minoration de l'entropie topologique des difféomorphismes du disque, J. London Math. Soc. 60 (1999), 912–924.
J. Franks, Knots, links and symbolic dynamics, Ann. of Math. 113 (1981), 529–552.
_____, Recurrence and fixed points of surface homeomorphisms, Ergodic Theory Dynam. Systems 8* (1988), 99–107.
_____, Realizing rotation vectors for torus homeomorphisms, Trans. Amer. Math. Soc. 311 (1989), 107–115.
_____, Geodesics on S 2 and periodic points of annulus homeomorphisms, Invent. Math. 108 (1992), 403–418.
J. Franks and M. Handel, Entropy and exponential growth of π 1 in dimension two, Proc. Amer. Math. Soc. 102 (1988), 753–760.
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.
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.
_____, Entropy and twisted cohomology, Topology 25 (1986), 455–470.
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.
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.
J. M. Gambaudo and J. Llibre, A note on the periods of surface homeomorphisms, J. Math. Anal. Appl. 177 (1993), 627–632.
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.
_____, Vers un ordre de Sarkovskii pour les plongements du disque préservant l'orientation, C. R. Acad. Sci. Paris, Sér. I 310 (1990), 291–294.
J. Guaschi, Lefschetz numbers of periodic orbits of pseudo-Anosov homeomorphisms, Math. Proc. Cambridge Philos. Soc. 115 (1994), 121–132.
_____, Pseudo-Anosov braid types of the disc or sphere of low cardinality imply all periods, J. London Math. Soc. 50 (1994), 594–608.
_____, Representations of Artin's braid groups and linking numbers of periodic orbits, J. Knot Theory Ramifications 4 (1995), 197–212.
_____, Nielsen theory, braids and fixed points of surface homeomorphisms, Topology Appl. 117 (2002), 199–230.
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.
G. R. Hall, A topological version of a theorem of Mather on twist maps, Ergodic Theory Dynam. Systems 4 (1984), 585–603.
T. Hall, Unremovable periodic orbits of homeomorphisms, Math. Proc. Cambridge Philos. Soc. 110 (1991), 523–531.
_____, Weak universality in two-dimensional transitions to chaos, Phys. Rev. Lett. 71 (1993), 58–61.
_____, Fat one-dimensional representatives of pseudo-Anosov isotopy classes with minimal periodic orbit structure, Nonlinearity 7 (1994), 367–384.
_____, The creation of horseshoes, Nonlinearity 7 (1994), 861–924.
M. Handel, The rotation set of a homeomorphism of the annulus is closed, Comm. Math. Phys. 127 (1990), 339–349.
_____, The forcing partial order on the three times punctured disk, Ergodic Theory Dynam. Systems 17 (1997), 593–610.
V. L. Hansen, Braids and Coverings: Selected Topics, London Math. Soc. Student Texts, vol. 18, Cambridge Univ. Press, Cambridge, 1989.
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.
_____, Markov maps on trees, Math. Japonica 51 (2000), 235–240.
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.
S. Husseini, Generalized Lefschetz numbers, Trans. Amer. Math. Soc. 272 (1982), 247–274.
B. Jiang, Lectures on Nielsen Fixed Point Theory, Contemp. Math., vol. 14, Amer. Math. Soc., Providence, 1983.
_____, A characterization of Nielsen classes, Fixed Point Theory and Its Applications, Contemp. Math., vol. 72, Amer. Math. Soc, Providence, 1988, pp. 157–160.
_____, Periodic orbits on surfaces via Nielsen fixed point theory, Topology-Hawaii (K. H. Dovermann, ed.), World Scientific, Singapore, 1991, pp. 101–118.
_____, 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.
_____, Estimation of the number of periodic orbits, Pacific J. Math. 172 (1996), 151–185.
_____, Bounds for fixed points on surfaces, Math. Ann. 311 (1998), 467–479.
B. Jiang and J. Guo, Fixed points of surface diffeomorphisms, Pacific J. Math. 160 (1993), 67–89.
M. R. Kelly, Computing Nielsen numbers of surface homeomorphisms, Topology 35 (1996), 13–25.
E. Kin, The forcing relation on periodic orbits of pseudo-Anosov braid types for disk automorphisms, preprint.
B. Kolev, Entropie topologique et représentation de Burau, C. R. Acad. Sci. Paris Sér. I 309 (1989), 835–838.
_____, Point fixe lié à une orbite périodique d'un difféomorphisme de \(\mathbb{R}^2 \) , C. R. Acad. Sci. Paris, Sér. 310 (1990), 831–833.
_____, Periodic orbits of period 3 in the disc, Nonlinearity 7 (1994), 1067–1071.
I. Kra, On the Nielsen-Thurston-Bers type of some self-maps of Riemann surfaces, Acta Math. 146 (1981), 231–270.
J. Kwapisz and R. Swanson, Asymptotic entropy, periodic orbits, and pseudo-Anosov maps, Ergodic Theory Dynam. Systems 18 (1998), 425–439.
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.
_____, Rotation vectors and entropy for homeomorphisms of the torus isotopic to the identity, Ergodic Theory Dynam. Systems 11 (1991), 115–128.
_____, Pseudo-Anosov homeomorphisms on a sphere with four punctures have all periods, Math. Proc. Cambridge Philos. Soc. 112 (1992), 539–549.
J. Los, Pseudo-Anosov maps and invariant train tracks in the disc: A finite algorithm, Proc. London Math. Soc. 66 (1993), 400–430.
_____, On the forcing relation for surface homeomorphisms, Publ. Math. IHES 85 (1997), 5–61.
S. Matsumoto, Rotation sets of surface homeomorphisms, Bol. Soc. Brasil. Mat. 28 (1997), 89–101.
_____, Arnold conjecture for surface homeomorphisms, Topology Appl. 104 (2000), 191–214.
T. Matsuoka, The number and linking of periodic solutions of periodic systems, Invent. Math. 70 (1983), 319–340.
_____, Waveform in the dynamical study of ordinary differential equations, Japan J. Appl. Math. 1 (1984), 417–434.
_____, 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.
_____, The number and linking of periodic solutions of non-dissipative systems, J. Differential Equations 76 (1988), 190–201.
_____, The number of periodic points of smooth maps, Ergodic Theory Dynam. Systems 9 (1989), 153–163.
_____, 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.
_____, Braid type and torsion number for fixed points of orientation-preserving embeddings on the disk, Math. Japonica 42 (1995), 25–34.
_____, Braid type of the fixed point set for orientation-preserving embeddings on the disk, Tokyo J. Math. 18 (1995), 457–472.
_____, Periodic points of disk homeomorphisms having a pseudo-Anosov component, Hokkaido Math. J. 27 (1998), 423–455.
_____, Braid invariants and instability of periodic solutions of time-periodic 2-dimensional ODE's, Topol. Methods Nonlinear Anal. 14 (1999), 261–274.
_____, On the linking structure of periodic orbits for embeddings of the disk, Math. Japonica 51 (2000), 241–254.
_____, Fixed point index and braid invariant for fixed points of embeddings on the disk, Topology Appl. 122 (2002), 337–352.
F. A. McRobie and J. M. T. Thompson, Braids and knots in driven oscillators, Internat. J. Bifur. Chaos 3 (1993), 1343–1361.
S. B. Miled and J. M. Gambaudo, Cascades of periodic orbits in two dimensions, Nonlinearity 10 (1997), 1627–1641.
M. Misiurewicz and K. Ziemian, Rotation sets for maps of tori, J. London Math. Soc. 40 (1989), 490–506.
_____, Rotation sets and ergodic measures for torus homeomorphisms, Fund. Math. 137 (1991), 45–52.
S. Moran, The Mathematical Theory of Knots and Braids: an Introduction, North-Holland Math. Studies, vol. 82, North-Holland, Amsterdam, 1983.
K. Murasugi, On closed 3-braids, Mem. Amer. Math. Soc. 151 (1974).
B. Peckham, The necessity of the Hopf bifurcation for periodically forced oscillators, Nonlinearity 3 (1990), 261–280.
M. Pollicott, Rotation sets for homeomorphisms and homology, Trans. Amer. Math. Soc. 331 (1992), 881–894.
M. Pollicott and R. Sharp, Growth of periodic points and rotation vectors on surfaces, Topology 36 (1997), 765–774.
D. Rolfsen, New developments in the theory of Artin's braid groups, Topology Appl. 127 (2003), 77–90.
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)
R. Sharp, Periodic points and rotation vectors for torus diffeomorphisms, Topology 34 (1995), 351–357.
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.
H. Shiraki, Fixed point indices of homeomorphisms defined on the torus, Hokkaido Math. J. 32 (2003), 59–74.
W. T. Song, K. H. Ko and J. Los, Entropies of braids, J. Knot Theory Ramifications 11 (2002), 647–666.
W. P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. 19 (1988), 417–431.
J. A. Walsh, Rotation vectors for toral maps and flows: a tutorial, Internat. J. Bifur. Chaos 5 (1995), 321–348.
Y. Yamaguchi and K. Tanikawa, Symmetrical non-Birkhoff period-3 orbits in standard-like mappings, Progr. Theoret. Phys. 104 (2000), 943–954.
_____, Non-symmetric non-Birkhoff period-2 orbits in the standard mapping, Progr. Theoret. Phys. 106 (2001), 691–696.
_____, Dynamical ordering of non-Birkhoff orbits and topological entropy in the standard mapping, Progr. Theoret. Phys. 107 (2002), 1117–1145.
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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
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