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A short treatise on the equivariant degree theory and its applications

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The aim of this survey is to give a profound introduction to equivariant degree theory, avoiding as far as possible technical details and highly theoretical background. We describe the equivariant degree and its relation to the Brouwer degree for several classes of symmetry groups, including also the equivariant gradient degree, and particularly emphasizing the algebraic, analytical, and topological tools for its effective calculation, the latter being illustrated by six concrete examples. The paper concludes with a brief sketch of the construction and interpretation of the equivariant degree.

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References

  1. Adams J.F.: Lectures on Lie Groups. W. A. Benjamin, Inc., New York (1969)

    MATH  Google Scholar 

  2. S. Antonian, An equivariant theory of retracts. In: Aspects of Topology, London Math. Soc. Lecture Note Ser. 93, Cambridge University Press, Cambridge, 1985, 251–269.

  3. J. Arpe, Berechnung sekundärer Koeffizientengruppen des SO(3) × S 1-äquivarianten Abbildungsgrades. Diploma thesis, University of Munich, 2001.

  4. Balanov Z., Farzamirad M., Krawcewicz W.: Symmetric systems of van der Pol equations. Topol. Methods Nonlinear Anal. 27, 29–90 (2006)

    MathSciNet  MATH  Google Scholar 

  5. Balanov Z., Farzamirad M., Krawcewicz W., Ruan H.: Applied equivariant degree, part II: Symmetric Hopf bifurcation for functional differential equations. Discrete Contin. Dyn. Syst. Ser. A 16, 923–960 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Balanov Z., Krawcewicz W.: Remarks on the equivariant degree theory. Topol. Methods Nonlinear Anal. 13, 91–103 (1999)

    MathSciNet  MATH  Google Scholar 

  7. Z. Balanov and W. Krawcewicz, Symmetric Hopf bifurcation: Twisted degree approach. In: Handbook of Differential Equations: Ordinary Differential Equations. Vol. 4, Elsevier, Amsterdam, 2008, 1–131.

  8. Balanov Z., Krawcewicz W., Ruan H.: Applied equivariant degree, part I: An axiomatic approach to primary degree. Discrete Contin. Dyn. Syst. Ser. A 15, 983–1016 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Balanov Z., Krawcewicz W., Ruan H.: Hopf bifurcation in a symmetric configuration of transmission lines. Nonlinear Anal. Real World Appl. 8, 1144–1170 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Z. Balanov, W. Krawcewicz and H. Ruan, Periodic solutions to O(2)-symmetric variational problems: O(2) × S 1-equivariant orthogonal degree approach. In: Proceedings of Conf. Nonlinear Analysis and Optimization (Haifa, 2008), Contemp. Math., American Mathematical Society, to appear.

  11. Balanov Z., Krawcewicz W., Steinlein H.: Reduced SO(3) × S 1-equivariant degree with applications to symmetric bifurcation problems. Nonlinear Anal. 47, 1617–1628 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. Balanov Z., Krawcewicz W., Steinlein H.: SO(3) × S 1-equivariant degree with applications to symmetric bifurcation problems: The case of one free parameter. Topol. Methods Nonlinear Anal. 20, 335–374 (2002)

    MathSciNet  MATH  Google Scholar 

  13. Z. Balanov, W. Krawcewicz and H. Steinlein, Applied Equivariant Degree. AIMS Series on Differential Equations & Dynamical Systems 1, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006.

  14. Balanov Z., Schwartzman E.: Morse complex, even functionals and asymptotically linear differential equations with resonance at infinity. Topol. Methods Nonlinear Anal. 12, 323–366 (1998)

    MathSciNet  MATH  Google Scholar 

  15. T. Bartsch, Topological Methods for Variational Problems with Symmetries. Lecture Notes in Math. 1560, Springer, Berlin, 1993.

  16. Bartsch T., Clapp M.: Critical point theory for indefinite functionals with symmetries. J. Funct. Anal. 138, 107–136 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  17. T. Bartsch and A. Szulkin, Hamiltonian systems: Periodic and homoclinic solutions by variational methods. In: Handbook of Differential Equations: Ordinary Differential Equations, Vol. 2, Elsevier, Amsterdam, 2005, 77–146.

  18. Benci V.: On critical point theory of indefinite functionals in the presence of symmetries. Trans. Amer. Math. Soc. 274, 533–572 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  19. Benci V., Rabinowitz P.: Critical point theorems for indefinite functionals. Invent. Math. 52, 241–273 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  20. Bredon G.E.: Introduction to Compact Transformation Groups. Academic Press, New York (1972)

    MATH  Google Scholar 

  21. Bröcker T., tom Dieck T.: Representations of Compact Lie Groups. Springer, New York (1985)

    MATH  Google Scholar 

  22. Buono P.-L., Golubitsky M.: Models of central pattern generators for quadruped locomotion. I. Primary gaits. J. Math. Biol. 42, 291–326 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  23. K. C. Chang, Infinite-Dimensional Morse Theory and Multiple Solution Problems. Progr. Nonlinear Differential Equations Appl. 6, Birkhäuser, Boston, 1993.

  24. Chossat P., Lauterbach R., Melbourne I.: Steady-state bifurcation with O(3)-symmetry. Arch. Ration. Mech. Anal. 113, 313–376 (1990)

    Article  MathSciNet  Google Scholar 

  25. Corbera M., Llibre J., Pérez-Chavela E.: Equilibrium points and central configurations for the Lennard-Jones 2- and 3-body problems. Celestial Mech. Dynam. Astronom. 89, 235–266 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  26. Costa D.G., Magalhães C.A.: A variational approach to subquadratic perturbations of elliptic systems. J. Differential Equations 111, 103–122 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  27. Costa D.G., Magalhães C.A.: A unified approach to a class of strongly indefinite functionals. J. Differential Equations 125, 521–547 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  28. Dancer E.N.: A new degree for S 1-invariant gradient mappings and applications. Ann. Inst. H. Poincaré Anal. Non Linéaire 2, 329–370 (1985)

    MathSciNet  MATH  Google Scholar 

  29. Dancer E.N.: Symmetries, degree, homotopy indices and asymptotically homogeneous problems. Nonlinear Anal. 6, 667–686 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  30. Dancer E.N., Gȩba K., Rybicki S.M.: Classification of homotopy classes of equivariant gradient maps. Fund. Math. 185, 1–18 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  31. Dancer E.N., Toland J.F.: The index change and global bifurcation for flows with a first integral. Proc. London Math. Soc. (3) 66, 539–567 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  32. T. tom Dieck, Transformation Groups and Representation Theory. Lecture Notes in Math. 766, Springer, Berlin, 1979.

  33. tom Dieck T.: Transformation Groups. Walter de Gruyter, Berlin (1987)

    MATH  Google Scholar 

  34. Dugundji J., Granas A.: Fixed Point Theory, Vol. I. PWN, Polish Scientific Publishers, Warsaw (1982)

    MATH  Google Scholar 

  35. Duistermaat J.J., Kolk J.A.C.: Lie Groups. Springer, Berlin (2000)

    MATH  Google Scholar 

  36. G. Dylawerski, An S 1-degree and S 1-maps between representation spheres. In: Algebraic Topology and Transformation Groups, T. tom Dieck (ed.), Lecture Notes in Math. 1361, Springer, Berlin, 1988, 14–28.

  37. Dylawerski G., Gȩba K., Jodel J., Marzantowicz W.: An S 1-equivariant degree and the Fuller index. Ann. Polon. Math. 52, 243–280 (1991)

    MathSciNet  MATH  Google Scholar 

  38. Erbe L.H., Krawcewicz W., Gȩba K., Wu J.: S 1-degree and global Hopf bifurcation theory of functional differential equations. J. Differential Equations 98, 277–298 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  39. Fang G.: Morse indices of degenerate critical orbits and applications—perturbation methods in equivariant cases. Nonlinear Anal. 36, 101–118 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  40. Fitzpatrick P.M., Pejsachowicz J., Recht L.: Spectral flow and bifurcation of critical points of strongly-indefinite functionals. I. General theory. J. Funct. Anal. 162, 52–95 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  41. J. Fura, A. Ratajczak and H. Ruan, Existence of nonstationary periodic solutions of Γ-symmetric asymptotically linear autonomous Newtonian systems with degeneracy. Rocky Mountain J. Math., to appear.

  42. Fura J., Ratajczak A., Rybicki S.: Existence and continuation of periodic solutions of autonomous Newtonian systems. J. Differential Equations 218, 216–252 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  43. Fura J., Rybicki S.: Periodic solutions of second order Hamiltonian systems bifurcating from infinity. Ann. Inst. H. Poincaré Anal. Non Linéaire 24, 471–490 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  44. K. Gȩba, Degree for gradient equivariant maps and equivariant Conley index. In: Topological Nonlinear Analysis, II (Frascati, 1995), Progr. Nonlinear Differential Equations Appl. 27, Birkhäuser, Boston, 1997, 247–272.

  45. Gȩba K., Izydorek M., Pruszko A.: The Conley index in Hilbert spaces and its applications. Studia Math. 134, 217–233 (1999)

    MathSciNet  Google Scholar 

  46. Gȩba K., Krawcewicz W., Wu J.H.: An equivariant degree with applications to symmetric bifurcation problems. I. Construction of the degree. Proc. London Math. Soc. 69, 377–398 (1994)

    Article  MathSciNet  Google Scholar 

  47. Gȩba K., Rybicki S.: Some remarks on the Euler ring U(G). J. Fixed Point Theory Appl. 3, 143–158 (2008)

    Article  MathSciNet  Google Scholar 

  48. A. Gołȩbiewska and S. Rybicki, Degree for invariant strongly indefinite functionals. Preprint.

  49. Golubitsky M., Stewart I.: Hopf bifurcation in the presence of symmetry. Arch. Rational Mech. Anal. 87, 107–165 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  50. M. Golubitsky and I. Stewart, Hopf bifurcation with dihedral group symmetry: Coupled nonlinear oscillators. In: Multiparameter Bifurcation Theory (Arcata, Calif., 1985), Contemp. Math. 56, American Mathematical Society, Providence, RI, 1986, 131–173.

  51. Golubitsky M., Stewart I., Buono P.-L., Collins J.J.: A modular network for legged locomotion. Phys. D 115, 56–72 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  52. Golubitsky M., Stewart I., Buono P.-L., Collins J.J.: Symmetry in locomotor central pattern generators and animal gaits. Nature 401, 693–695 (1999)

    Article  Google Scholar 

  53. M. Golubitsky, I. N. Stewart and D. G. Schaeffer, Singularities and Groups in Bifurcation Theory, Vol. II. Applied Mathematical Sciences 69, Springer, New York, 1988.

  54. Guo Z., Yu J.: Multiplicity results for periodic solutions to delay differential equations via critical point theory. J. Differential Equations 218, 15–35 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  55. Ihrig E., Golubitsky M.: Pattern selection with O(3)-symmetry. Phys. D 13, 1–33 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  56. J. Ize, Topological bifurcation. In: Topological Nonlinear Analysis, Progr. Nonlinear Differential Equations Appl. 15, Birkhäuser, Boston, 1995, 341–463.

  57. J. Ize, Equivariant degree. In: Handbook of Topological Fixed Point Theory, Springer, Dordrecht, 2005, 331–337.

  58. Ize J., Massabò I., Vignoli A.: Degree theory for equivariant maps. I. Trans. Amer. Math. Soc. 315, 433–510 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  59. J. Ize, I. Massabò and A. Vignoli, Degree theory for equivariant maps, the general S 1-action. Mem. Amer. Math. Soc. 100 (1992), no. 481.

  60. J. Ize and A. Vignoli, Equivariant Degree Theory. De Gruyter Series in Nonlinear Analysis and Applications 8, Walter de Gruyter, Berlin, 2003.

  61. Izydorek M.: Equivariant Conley index in Hilbert spaces and applications to strongly indefinite problems. Nonlinear Anal. 51, 33–66 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  62. Kawakubo K.: The Theory of Transformation Groups. The Clarendon Press, Oxford University Press, New York (1991)

    MATH  Google Scholar 

  63. Komiya K.: The Lefschetz number for equivariant maps. Osaka J. Math. 24, 299–305 (1987)

    MathSciNet  MATH  Google Scholar 

  64. Kosniowski C.: Equivariant cohomology and stable cohomotopy. Math. Ann. 210, 83–104 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  65. M. A. Krasnosel’skiĭ and P. P. Zabreĭko, Geometrical Methods of Nonlinear Analysis. Grundlehren der mathematischen Wissenschaften 263, Springer, Berlin, 1984.

  66. Krawcewicz W., Vivi P., Wu J.: Computational formulae of an equivariant degree with applications to symmetric bifurcations. Nonlinear Stud. 4, 89–119 (1997)

    MathSciNet  MATH  Google Scholar 

  67. Krawcewicz W., Vivi P., Wu J.: Hopf bifurcations of functional differential equations with dihedral symmetries. J. Differential Equations 146, 157–184 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  68. W. Krawcewicz and J. Wu, Theory of Degrees with Applications to Bifurcations and Differential Equations. Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, New York, 1997.

  69. Krawcewicz W., Wu J.: Theory and applications of Hopf bifurcations in symmetric functional-differential equations. Nonlinear Anal. 35, 845–870 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  70. W. Krawcewicz, J. Wu and H. Xia, Global Hopf bifurcation theory for condensing fields and neutral equations with applications to lossless transmission problems. Canad. Appl. Math. Quart. 1 (1993), 167–220.

    Google Scholar 

  71. Kryszewski W., Szulkin A.: An infinite-dimensional Morse theory with applications. Trans. Amer. Math. Soc. 349, 3181–3234 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  72. A. Kushkuley and Z. Balanov, Geometric Methods in Degree Theory for Equivariant Maps. Lecture Notes in Math. 1632, Springer, Berlin, 1996.

  73. L. G. Lewis Jr., J. P. May, M. Steinberger and J. E. McClure, Equivariant Stable Homotopy Theory. Lecture Notes in Math. 1213, Springer, Berlin, 1986.

  74. W. Lück, The equivariant degree. In: Algebraic Topology and Transformation Groups (Göttingen, 1987), Lecture Notes in Math. 1361, Springer, Berlin, 1988, 123–166.

  75. Marzantowicz W., Prieto C.: Computation of the equivariant 1-stem. Nonlinear Anal. 63, 513–524 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  76. Marzantowicz W., Prieto C., Rybicki S.: Periodic solutions of symmetric autonomous Newtonian systems. J. Differential Equations 244, 916–944 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  77. Y. Matsumoto, An Introduction to Morse Theory. Translations of Mathematical Monographs 208, American Mathematical Society, Providence, RI, 2002.

  78. J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems. Applied Mathematical Sciences 74, Springer, New York, 1989.

  79. Mayer K.H.: G-invariante Morse-Funktionen. Manuscripta Math. 63, 99–114 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  80. Nagumo M.: Degree of mapping in convex linear topological spaces. Amer. J. Math. 73, 497–511 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  81. Namboodiri U.: Equivariant vector fields on spheres. Trans. Amer. Math. Soc. 278, 431–460 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  82. Palis J., de Melo W.: Geometric Theory of Dynamical Systems. An Introduction. Springer, New York (1982)

    MATH  Google Scholar 

  83. Parusiński A.: Gradient homotopies of gradient vector fields. Studia Math. 96, 73–80 (1990)

    MathSciNet  MATH  Google Scholar 

  84. Pinto C.M.A., Golubitsky M.: Central pattern generators for bipedal locomotion. J. Math. Biol. 53, 474–489 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  85. Rabinowitz P.H.: Some global results for nonlinear eigenvalue problems. J. Funct. Anal. 7, 487–513 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  86. Rabinowitz P.H.: Periodic solutions of Hamiltonian systems. Comm. Pure Appl. Math. 31, 157–184 (1978)

    Article  MathSciNet  Google Scholar 

  87. P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Regional Conference Series in Mathematics 65, American Mathematical Society, Providence, RI, 1986.

  88. Radzki W.: Degenerate branching points of autonomous Hamiltonian systems. Nonlinear Anal. 55, 153–166 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  89. Radzki W., Rybicki S.: Degenerate bifurcation points of periodic solutions of autonomous Hamiltonian systems. J. Differential Equations 202, 284–305 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  90. Ruan H., Rybicki S.: Applications of equivariant degree for gradient maps to symmetric Newtonian systems. Nonlinear Anal. 68, 1479–1516 (2008)

    MathSciNet  MATH  Google Scholar 

  91. Y. Rudyak, On Thom spectra, Orientability, and Cobordism. Corr. 2nd printing, Springer, Berlin, 2007.

  92. S. Rybicki, A degree for S 1-equivariant orthogonal maps and its applications to bifurcation theory. Nonlinear Anal. 23 (1994), 83–102.

    Google Scholar 

  93. Rybicki S.: Applications of degree for S 1-equivariant gradient maps to variational nonlinear problems with S 1-symmetries. Topol. Methods Nonlinear Anal. 9, 383–417 (1997)

    MathSciNet  MATH  Google Scholar 

  94. Rybicki S.: Degree for S 1-equivariant strongly indefinite functionals. Nonlinear Anal. 43, 1001–1017 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  95. S. Rybicki, Bifurcations of solutions of SO(2)-symmetric nonlinear problems with variational structure. In: Handbook of Topological Fixed Point Theory, R. Brown, M. Furi, L. Górniewicz and B. Jiang (eds.), Springer, Dordrecht, 2005, 339–372.

  96. Rybicki S.: Degree for equivariant gradient maps. Milan J. Math. 73, 103–144 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  97. Spanier E.H.: Algebraic Topology. McGraw-Hill, New York (1966)

    MATH  Google Scholar 

  98. H. Steinlein, Borsuk’s antipodal theorem and its generalizations and applications: A survey. In: Méthodes topologiques en analyse non linéaire, Sém. Math. Sup. 95, Presses Univ. Montréal, Montreal, QC, 1985, 166–235.

  99. R. E. Stong, Notes on Cobordism Theory. Princeton University Press; University of Tokyo Press, 1968.

  100. H. Ulrich, Fixed Point Theory of Parametrized Equivariant maps. Lecture Notes in Math. 1343, Springer, Berlin, 1988.

  101. Wasserman A.G.: Equivariant differential topology. Topology 8, 127–150 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  102. J. Wu, Theory and Applications of Partial Functional-Differential Equations. Applied Mathematical Sciences 119, Springer, New York, 1996.

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Balanov, Z., Krawcewicz, W., Rybicki, S. et al. A short treatise on the equivariant degree theory and its applications. J. Fixed Point Theory Appl. 8, 1–74 (2010). https://doi.org/10.1007/s11784-010-0033-9

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