Symmetry breaking for compact Lie groups

Memoirs of the American Mathematical Society

Volumn , Issue 574, 1996, Pages 1-170

  Field, Michael ^a  

^a NONE

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EID: 0030101770     PISSN: 00659266     EISSN: None     Source Type: Journal    
DOI: 10.1090/memo/0574     Document Type: Article

References (79)

1. [1] R. Abraham and J. W. Robbin. Transversal Mappings and Flows, (Benjamin, New York, 1967).
2. [2] K. Aoki, T. Fukada and T. Nishimura. 'An algebraic formula for the topological type of one parameter bifurcation diagrams', Archive Rational Mech. and Anal., 108 (1989), 247-266.
3. N symmetry', Nonlinearity, 4 (1991), 861-902.
4. [4] M. Artin. 'On the Solutions of Analytic Equations', Invent. math., 5,(1968), 277-291.
5. [5] E. Bierstone. 'Lifting isotopies from orbit spaces', Topology, 14 (1975), 245-252.
6. [6] E. Bierstone. 'General position of equivariant maps', Trans. Amer. Math. Soc., 234 (1977), 447-466.
7. [7] E. Bierstone. 'Generic equivariant maps', Real and Complex Singularities, Oslo 1976, Proc. Nordic Summer School/NAVF Sympos. Math. (Sijthoff and Noordhoff International Publ.) Leyden (1977), 127-161.
8. [8] E. Bierstone and P. D. Milman. 'Semianalytic and subanalytic sets', Publ. I.H.E.S., 67 (1988), 5-42.
9. [9] E. Bierstone and P. D. Milman. 'Uniformization of analytic spaces', Jnr. Amer. Math. Soc., 2(4) (1989), 801-836.
10. [10] E. Bierstone and P. D. Milman. 'A simple constructive proof of canonical resolution of singularities', Effective methods in algebraic geometry, (Castiglioncello, 1990), 11-30 (Progr. Math. 94, Birkaüser, Boston.)
11. [10] E. Bierstone and P. D. Milman. 'A simple constructive proof of canonical resolution of singularities', Effective methods in algebraic geometry, (Castiglioncello, 1990), 11-30 (Progr. Math. 94, Birkaüser, Boston.)
12. [11] E. Bierstone and P. D. Milman. 'Canonical desingularization in characteristic zero: a simple constructive proof', in preparation.
13. [12] J. Bochnak, M. Coste and M-F. Roy. Géométrie algébrique réelle (Ergebnisse der Mathematik und ihrer Grenzgebiete; Folge 3, Bd. 2, Springer-Verlag, Berlin, Heidelberg, New York, 1987.)
14. [13] G. E. Bredon. Introduction to compact transformation groups (Pure and Applied Mathematics, 46, Academic Press, New York and London, 1972.)
15. [14] E. Brieskorm and H. Knörrer. Plane Algebraic Curves, (Birkhäuser Verlag Basel, Boston, Stuttgart, 1986.)
16. [15] T. Bröcker and T. tom Dieck. Representations of Compact Lie Groups, (Graduate Texts in Mathematics, Springer, New York, 1985).
17. [16] H. Cartan. 'Variétés analytiques réelles et variétés analytiques complexes', Bull. Soc. math France, 85 (1957), 77-99.
18. [17] P. Chossat and M. J. Field. 'Geometric analysis of the effect of symmetry breaking perturbations on an O(2) invariant homoclinic cycle', to appear in Fields Institute Communications.
19. [18] P. Chossat, R. Lauterbach and I. Melbourne. 'Steady-state bifurcation with O(3)-symmetry', Arch. Rational Mech. Anal., 113 (4), (1990), 313-376.
20. [19] G. Cicogna and G. Gaeta. 'Quaternionic-like bifurcation in the absence of symmetry', J. Phys. A: Math. Gen. 20 (1987), 79-89.
21. [20] M. Coste. 'Ensembles semi-algébriques', in Géométrie Algebébrique Réelle et Formes Quadratiques, Springer Lecture Notes in Math., 959, 1982, 109-138.
22. [21] J. Damon. 'Topological equivalence of bifurcation problems', Nonlinearity, 1 (1988), 311-331.
23. [22] J. Damon. 'Equivariant bifurcations and Morseifications for finite groups', in Singularity Theory and its Applications, II, eds. M. Roberts and I. Stewart, Springer Lecture Notes in Math., 1463 (1991), 80-106.
24. [23] J. Damon. 'G-signature, G-degree, and Symmetries of the Branches of Curve Singularities', Topology, 30 (1991), 565-590.
25. [24] B. Fiedler. Global Bifurcation of Periodic Solutions with Symmetry, (Springer Lecture Notes in Math., 1309, Springer-verlag, New York-London, 1988.)
26. [25] M. J. Field. 'Transversality in G-manifolds', Trans. Amer. Math. Soc., 231 (1977), 429-450.
27. [26] M. J. Field. 'Equivariant Dynamical Systems', Trans. Amer. Math. Soc., 259 (1980),185-205.
28. [27] M. J. Field. 'Resolving actions of compact Lie groups', Bull. Austral. Math. Soc. 18 (1978), 243-254.
29. [28] M. J. Field. 'Equivariant Dynamics', Contemp. Math, 56 (1986), 69-95.
30. [29] M. J. Field. 'Equivariant Bifurcation Theory and Symmetry Breaking', J. Dynamics and Diff. Eqns., 1(4) (1989), 369-421.
31. [30] M. J. Field. 'Local structure of equivariant dynamics', in Singularity Theory and its Applications, II, eds. M. Roberts and I. Stewart, Springer Lecture Notes in Math., 1463 (1991), 168-195.
32. [31] M. J. Field, 'Determinacy and branching patterns for the equivariant Hopf bifurcation', Nonlinearity, 7(1994), 403-415.
33. [32] M. J. Field, 'Geometric methods in bifurcation theory', In: Pattern formation and symmetry breaking in PDEs. Fields Institute Communications, AMS, to appear.
34. [33] M. J. Field, 'Blowing-up in equivariant bifurcation theory', in Dynamics, Bifurcation and Symmetries: New Trends and New Tools (P. Chossat and J.-M. Gambaudo, Eds) NATO ARW Series, Kluwer, Amsterdam (1994), 111-122.
35. [34] M. J. Field. 'Symmetry breaking for equivariant maps', preprint, University of Houston (1994).
36. [35] M. J. Field and R. W. Richardson. 'Symmetry Breaking and the Maximal Isotropy Subgroup Conjecture for Reflection Groups', Arch. for Rational Mech. and Anal., 105(1) (1989), 61-94.
37. [36] M. J. Field and R. W. Richardson. 'Symmetry breaking in equivariant bifurcation problems', Bull. Amer. Math. Soc., 22(1) (1990), 79-84.
38. [37] M. J. Field and R. W. Richardson. 'Symmetry breaking and branching patterns in equivariant bifurcation theory I', Arch. Rational Mech. Anal. 118 (1992), 297-348.
39. [38] M. J. Field and R. W. Richardson. 'Symmetry breaking and branching patterns in equivariant bifurcation theory II', Arch. Rational Mech. Anal. 120 (1992), 147-190.
40. [39] M. J. Field and J. W. Swift. 'The Hopf bifurcation and the Hopf fibration', Nonlinearity, 7 (1994), 385-402.
41. [40] K. Gatermann. 'Semi-Invariants, Equivariants and Algorithms', Applicable Algebra in Engineering, Communication and Computing, to appear.
42. [41] C. Gibson, K. Wirthmüller, A. A. du Plessis and E. Looijenga. Topological Stability of smooth mappings, (Springer Lecture Notes in Math. 553, 1976).
43. [42] M. Golubitsky and D. G. Schaeffer. Singularities and Groups in Bifurcation Theory, Vol. I, (Appl. Math. Sci. 51, Springer-Verlag, New York, 1985).
44. [43] M. Golubitsky, I. N. Stewart and D. G. Schaeffer. Singularities and Groups in Bifurcation Theory, Vol. II, (Appl. Math. Sci. 69, Springer-Verlag, New York, 1988).
45. [44] M. Golubitsky and I. N. Stewart. 'Hopf bifurcation in the presence of symmetry', Arch. for Rational Mech. and Anal. 87 (1985), 107-165.
46. [45] J. Guckenheimer and P. Holmes. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, (Applied Mathematical Sciences 42, Springer-Verlag, New York, 1983).
47. [46] S. Helgason. Differential Geometry, Lie Groups, and Symmetric Spaces, (Academic Press, New York, San Francisco, London, 1978.)
48. [47] H. Hironaka. 'Resolution of singularities of an algebraic variety over a field of characteristic zero:I, II', Ann. of Math. (2), 79 (1964), 109-326.
49. [48] H. Hironaka. 'Subanalytic sets', Number Theory, Algebraic Geometry and Commutative Algebra, Tokyo, Kinokuniya, 1973, 453-493.
50. [49] M. W. Hirsch, C. C. Pugh and M. Shub. Invariant Manifolds, (Springer Lect. Notes Math., 583, 1977).
51. [50] J. Ize. Private communication.
52. [51] N. Jacobson. Basic Algebra I, (W. H. Freeman, San Francisco, 1974.)
53. [52] M. Krupa. 'Bifurcations of relative equilibria', SIAM J. MATH. ANAL., 21(6) (1990), 1453-1486.
54. [53] S. G. Krantz and H. R. Parks. A Primer of Real Analytic Functions, (Birkhäser, Basel, Boston, Berlin, 1992).
55. [54] T. C. Kuo. 'The Ratio test for Analytic Whitney Stratifications', Proc. of Liverpool Sing. Symp. Vol. 1, Springer Lecture Notes in Math., 192 (1971), 141-149.
56. [55] T. C. Kuo. 'On analytic function germs of two complex variables', Topology, 16 (1977), 299-310.
57. [56] T. C. Kuo. 'The modified analytic trivialization of singularities', J. Math. Soc. Japan, 32(4), (1980), 605-614.
58. [57] T. C. Kuo. 'On classification of real singularities', Invent. Math., 82 (1985), 257-262.
59. [58] R. Lauterbach and M. Roberts. 'Heteroclinic cycles in Dynamical systems with broken spherical symmetry', J. Diff. Eq., 100(1) (1992), 22-48.
60. [59] J. Mather. 'Stratifications and mappings', Proceedings of the Dynamical Systems Conference, Salvador, Brazil, ed. M. Peixoto, (Academic Press, New-York, San Franscisco, London, 1973.)
61. [60] J. Mather. 'Differentiable invariants', Topology, 16 (1977), 145-155.
62. [61] T. Matumoto and M. Shiota. 'Unique triangulation of the orbit space of a differentiable transformation group and its applications', Homotopy theory and related topics, (Kyoto, 1984), 41-55, (Adv. Stud. Pure Math. 9, North-Holland, Amsterdam-New York, 1987).
63. [62] I. Melbourne. 'Maximal isotropy subgroups for absolutely irreducible representations of compact Lie groups', Nonlinearity 7(5) (1994), 1385-1394.
64. [63] J. Milnor. Singular points of complex hypersurfaces,(Ann. of Math. Studies 61, Princeton Univ. Press, 1968.)
65. ∞ en Présence de Symétrie, (Springer Lect. Notes in Math. 510), Springer-Verlag, New York, 1976.)
66. [65] M. Roberts. 'On the genericity of some properties of equivariant map germs', J. London Math. Soc. 32(2) (1985), 177-192.
67. [66] D. Ruelle, 'Bifurcation in the presence of a symmetry group', Arch. Rational Mech. Anal., 51(2) (1973), 136-152.
68. [67] G. W. Schwarz. 'Smooth functions invariant under the action of a compact Lie group', Topology, 14 (1975), 63-68.
69. [68] G. W. Schwarz. 'Lifting smooth isotopies of orbit spaces', Publ. I.H.E.S., 51 (1980), 37-135.
70. [69] G. W. Schwarz. 'Algebraic quotients of compact group actions', preprint (1994).
71. [70] R. P. Stanley. 'Invariants of finite groups and their applications to combinatorics', Bull. Amer. Math. Soc, 1 (1979), 475-511.
72. [71] I. N. Stewart. 'Stability of periodic solutions in symmetric Hopf bifurcation', Dynamics and Stability of Systems, 2(4) (1988), 149-165.
73. [72] J. W. Swift. 'Hopf bifurcation with the symmetry of the square', Nonlinearity 1 (1988), 333-377.
74. [73] S. D. Taliaferro. 'Stability of bifurcating solutions in the presence of symmetry', Nonlinear Analysis Theory Methods and Applications, 8 (1987), 821-839.
75. [74] J. C. Tougeron. Ideaux de fonctions differentiable, (Erge. der Math, und ihrer Gren., 71, Springer-Verlag, Berlin, Heidelberg, New-York, 1972.)
76. [75] B. L. van der Waerden. Algebra, Volume 1 (Seventh Edition), (Frederick Unger, New York,1977.)
77. [76] R. J. Walker. Algebraic Curves, (Princeton University Press, 1950.)
78. [77] H. Whitney Complex Analytic Varieties, (Adison-Wesley, Reading - Massachusetts, Menlo Park - California, London, Don Mills - Ontario, 1972.)
79. [78] P. A. Worfolk. 'Zeros of equivariant vector fields: Algorithms for an invariant approach', JSC 17 (1994), 487-511.