April  2013, 6(4): 985-997. doi: 10.3934/dcdss.2013.6.985

Equivariant Conley index versus degree for equivariant gradient maps

1. 

Faculty of Mathematics and Computer Science

2. 

Nicolaus Copernicus University

3. 

ul. Chopina 12/18, PL-87-100 Toru?

Received  October 2011 Revised  April 2012 Published  December 2012

In this article we study the relationship between the degree forinvariant strongly indefinite functionals and the equivariantConley index. We prove that, under certain assumptions, achange of the equivariant Conley indices is equivalent to thechange of the degrees for equivariant gradient maps. Moreover, weformulate easy to verify sufficient conditions for theexistence of a global bifurcation of critical orbits of invariantstrongly indefinite functionals.
Citation: Anna Go??biewska, S?awomir Rybicki. Equivariant Conley index versus degree for equivariant gradient maps. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 985-997. doi: 10.3934/dcdss.2013.6.985
References:
[1]

A. Ambrosetti, Branching points for a class of variational operators,, J. Anal. Math., 76 (1998), 321. doi: 10.1007/BF02786940.

[2]

Z. Balanov, W. Krawcewicz, S. Rybicki and H. Steinlein, A short treatise on the equivariant degree theory and its applications,, J. Fixed Point Theory App., 8 (2010), 1. doi: 10.1007/s11784-010-0033-9.

[3]

Z. Balanov, W. Krawcewicz and H. Ruan, Periodic solutions to $O(2)$-symmetric variational problems: $O(2)\times S^1$-equivariant gradient degree approach,, Israel Math. Conf. Proc., 514 (2008), 45. doi: 10.1090/conm/514/10099.

[4]

A. Banyaga and D. Hurtubise, "Lectures on Morse Homology,", Kluwer Academic Publishers, (2004).

[5]

P. Bartłomiejczyk. K. Gęba and M. Izydorek, Otopy classes of equivariant maps,, J. Fixed P. Th. and Appl., 7 (2010), 145. doi: 10.1007/s11784-010-0013-0.

[6]

T. Bartsch, "Topological Methods for Variational Problems with Symmetries,", Lect. Notes in Math., 1560 (1993).

[7]

R. Böhme, Die lösung der versweigungsgleichungen für nichtlineare eigenwert-probleme,, Math. Z., 127 (1972), 105.

[8]

G. Bredon, "Introduction to Compact Transformation Groups,", Academic Press, (1972).

[9]

S. N. Chow and R. Lauterbach, A bifurcation theorem for critical points of variational problems,, Nonl. Anal. TMA, 12 (1988), 51. doi: 10.1016/0362-546X(88)90012-0.

[10]

Ch. Conley, "Isolated Invariant Sets and the Morse Index,", CBMS Regional Conference Series in Mathematics, 38 (1978).

[11]

E. N. Dancer, A new degree for $S^1$-invariant mappings and applications,, Ann. Inst. H.Poincaré, 2 (1985), 329.

[12]

E. N. Dancer, K. Gęba and S. Rybicki, Classification of homotopy classes of equivariant gradient maps,, Fund. Math., 185 (2005), 1. doi: 10.4064/fm185-1-1.

[13]

T. tom Dieck, "Transformation Groups and Representation Theory,", Springer-Verlag, (1979).

[14]

T. tom Dieck, "Transformation Groups,", Walter de Gruyter, (1987). doi: 10.1515/9783110858372.312.

[15]

J. J. Duistermaat and J. A. C. Kolk, "Lie Groups,", Springer-Verlag, (2000). doi: 10.1007/978-3-642-56936-4.

[16]

G. Fang, Morse indices of degenerate critical orbits and applications - perturbation methods in equivariant cases,, Nonl. Anal. TMA, 36 (1999), 101. doi: 10.1016/S0362-546X(98)00017-0.

[17]

A. Floer, A refinement of the Conley index and an application to the stability of hyperbolic invariant sets,, Erg. Th. and Dynam. Sys., 7 (1987), 93. doi: 10.1017/S0143385700003825.

[18]

K. Gęba, Degree for gradient equivariant maps and equivariant Conley index,, Topological Nonlinear Analysis, 27 (1997), 247.

[19]

K. Gęba, M. Izydorek and A. Pruszko, The Conley index in Hilbert spaces and its applications,, Studia Math., 134 (1999), 217.

[20]

A. Gołębiewska and S. Rybicki, Global bifurcations of critical orbits of $G$-invariant strongly indefinite functionals,, Nonl. Anal TMA, 74 (2011), 1823. doi: 10.1016/j.na.2010.10.055.

[21]

J. Ize, "Bifurcation Theory for Fredholm Operators,", Mem. AMS, 174 (1976).

[22]

J. Ize, Topological bifurcation,, Topological Nonlinear Analysis, 15 (1995), 341.

[23]

M. Izydorek, A Cohomological Conley index in Hilbert spaces and applications to strongly indefinite problems,, J. Diff. Equat., 170 (2001), 22. doi: 10.1006/jdeq.2000.3818.

[24]

M. Izydorek, Equivariant Conley index in Hilbert spaces and applications to strongly indefinite problems,, Nonl. Anal. TMA, 51 (2002), 33. doi: 10.1016/S0362-546X(01)00811-2.

[25]

G. López Garza and S. Rybicki, Equivariant bifurcation index,, Nonl. Anal. TMA, 73 (2010), 2779. doi: 10.1016/j.na.2010.06.001.

[26]

A. Marino, La biforcazione nel caso variazionale,, Conf. Sem. Mat. Univ. Bari, 132 (1973).

[27]

K. H. Mayer, G-invariante Morse-funktionen,, Man. Math., 63 (1989), 99. doi: 10.1007/BF01173705.

[28]

K. Mischaikow and M. Mrozek, Conley index,, Handbook of Dynamical Systems, 2 (2002), 393. doi: 10.1016/S1874-575X(02)80030-3.

[29]

L. Nirenberg, "Topics in Nonlinear Functional Analysis,", Courant Institute of Mathematical Sciences, (1974).

[30]

P. H. Rabinowitz, Nonlinear Sturm-Liouville problems for second order ordinary differential equations,, Comm. Pure Appl. Math., 23 (1970), 939.

[31]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems,, J. Func. Anal., 7 (1971), 487.

[32]

P. H. Rabinowitz, A global theorem for nonlinear eigenvalue problems and applications,, Contributions to Nonlinear Functional Analysis, (1971), 11.

[33]

W. Radzki and S. Rybicki, Degenerate bifurcation points of periodic solutions of autonomous Hamiltonian systems,, J. Diff. Equat., 202 (2004), 284. doi: 10.1016/j.jde.2004.03.037.

[34]

H. Ruan and S. Rybicki, Applications of equivariant degree for gradient maps to symmetric Newtonian systems,, Nonl. Anal. TMA, 68 (2008), 1479. doi: 10.1016/j.na.2006.12.039.

[35]

K. Rybakowski, On the homotopy index for infinite dimensional semiflows,, Trans. Amer. Math. Soc., 269 (1982), 351. doi: 10.2307/1998452.

[36]

S. Rybicki, A degree for $S^1$-equivariant orthogonal maps and its applications to bifurcation theory,, Nonl. Anal. TMA, 23 (1994), 83. doi: 10.1016/0362-546X(94)90253-4.

[37]

S. Rybicki, Applications of degree for $S^1$-equivariant gradient maps to variational nonlinear problems with $S^1$-symmetries,, Topol. Meth. Nonl. Anal., 9 (1997), 383.

[38]

S. Rybicki, Degree for equivariant gradient maps,, Milan J. Math., 73 (2005), 103. doi: 10.1007/s00032-005-0040-2.

[39]

S. Rybicki, Global bifurcations of critical orbits via equivariant Conley index,, Adv. Nonl. Stud., 11 (2011), 929.

[40]

D. Salamon, Connected simple systems and the Conley index of isolated invariant sets,, TAMS, 291 (1985), 1. doi: 10.2307/1999893.

[41]

D. Salamon, Morse theory, the Conley index and Floer homology,, Bull. London Math. Soc., 22 (1990), 113. doi: 10.1112/blms/22.2.113.

[42]

J. Smoller and A. Wasserman, Bifurcation and symmetry-breaking,, Invent. Math., 100 (1990), 63. doi: 10.1007/BF01231181.

[43]

J. Smoller, "Shock Waves and Reaction-diffusion Equations,", Fundamental Principles of Mathematical Science, 258 (1983).

[44]

F. Takens, Some remarks on the Böhme-Berger bifurcation theorem,, Math. Z., 125 (1972), 359.

[45]

G. W. Whitehead, "Recent Advances in Homotopy Theory,", CMBS Regional Conf. Ser. in Math., 5 (1970).

show all references

References:
[1]

A. Ambrosetti, Branching points for a class of variational operators,, J. Anal. Math., 76 (1998), 321. doi: 10.1007/BF02786940.

[2]

Z. Balanov, W. Krawcewicz, S. Rybicki and H. Steinlein, A short treatise on the equivariant degree theory and its applications,, J. Fixed Point Theory App., 8 (2010), 1. doi: 10.1007/s11784-010-0033-9.

[3]

Z. Balanov, W. Krawcewicz and H. Ruan, Periodic solutions to $O(2)$-symmetric variational problems: $O(2)\times S^1$-equivariant gradient degree approach,, Israel Math. Conf. Proc., 514 (2008), 45. doi: 10.1090/conm/514/10099.

[4]

A. Banyaga and D. Hurtubise, "Lectures on Morse Homology,", Kluwer Academic Publishers, (2004).

[5]

P. Bartłomiejczyk. K. Gęba and M. Izydorek, Otopy classes of equivariant maps,, J. Fixed P. Th. and Appl., 7 (2010), 145. doi: 10.1007/s11784-010-0013-0.

[6]

T. Bartsch, "Topological Methods for Variational Problems with Symmetries,", Lect. Notes in Math., 1560 (1993).

[7]

R. Böhme, Die lösung der versweigungsgleichungen für nichtlineare eigenwert-probleme,, Math. Z., 127 (1972), 105.

[8]

G. Bredon, "Introduction to Compact Transformation Groups,", Academic Press, (1972).

[9]

S. N. Chow and R. Lauterbach, A bifurcation theorem for critical points of variational problems,, Nonl. Anal. TMA, 12 (1988), 51. doi: 10.1016/0362-546X(88)90012-0.

[10]

Ch. Conley, "Isolated Invariant Sets and the Morse Index,", CBMS Regional Conference Series in Mathematics, 38 (1978).

[11]

E. N. Dancer, A new degree for $S^1$-invariant mappings and applications,, Ann. Inst. H.Poincaré, 2 (1985), 329.

[12]

E. N. Dancer, K. Gęba and S. Rybicki, Classification of homotopy classes of equivariant gradient maps,, Fund. Math., 185 (2005), 1. doi: 10.4064/fm185-1-1.

[13]

T. tom Dieck, "Transformation Groups and Representation Theory,", Springer-Verlag, (1979).

[14]

T. tom Dieck, "Transformation Groups,", Walter de Gruyter, (1987). doi: 10.1515/9783110858372.312.

[15]

J. J. Duistermaat and J. A. C. Kolk, "Lie Groups,", Springer-Verlag, (2000). doi: 10.1007/978-3-642-56936-4.

[16]

G. Fang, Morse indices of degenerate critical orbits and applications - perturbation methods in equivariant cases,, Nonl. Anal. TMA, 36 (1999), 101. doi: 10.1016/S0362-546X(98)00017-0.

[17]

A. Floer, A refinement of the Conley index and an application to the stability of hyperbolic invariant sets,, Erg. Th. and Dynam. Sys., 7 (1987), 93. doi: 10.1017/S0143385700003825.

[18]

K. Gęba, Degree for gradient equivariant maps and equivariant Conley index,, Topological Nonlinear Analysis, 27 (1997), 247.

[19]

K. Gęba, M. Izydorek and A. Pruszko, The Conley index in Hilbert spaces and its applications,, Studia Math., 134 (1999), 217.

[20]

A. Gołębiewska and S. Rybicki, Global bifurcations of critical orbits of $G$-invariant strongly indefinite functionals,, Nonl. Anal TMA, 74 (2011), 1823. doi: 10.1016/j.na.2010.10.055.

[21]

J. Ize, "Bifurcation Theory for Fredholm Operators,", Mem. AMS, 174 (1976).

[22]

J. Ize, Topological bifurcation,, Topological Nonlinear Analysis, 15 (1995), 341.

[23]

M. Izydorek, A Cohomological Conley index in Hilbert spaces and applications to strongly indefinite problems,, J. Diff. Equat., 170 (2001), 22. doi: 10.1006/jdeq.2000.3818.

[24]

M. Izydorek, Equivariant Conley index in Hilbert spaces and applications to strongly indefinite problems,, Nonl. Anal. TMA, 51 (2002), 33. doi: 10.1016/S0362-546X(01)00811-2.

[25]

G. López Garza and S. Rybicki, Equivariant bifurcation index,, Nonl. Anal. TMA, 73 (2010), 2779. doi: 10.1016/j.na.2010.06.001.

[26]

A. Marino, La biforcazione nel caso variazionale,, Conf. Sem. Mat. Univ. Bari, 132 (1973).

[27]

K. H. Mayer, G-invariante Morse-funktionen,, Man. Math., 63 (1989), 99. doi: 10.1007/BF01173705.

[28]

K. Mischaikow and M. Mrozek, Conley index,, Handbook of Dynamical Systems, 2 (2002), 393. doi: 10.1016/S1874-575X(02)80030-3.

[29]

L. Nirenberg, "Topics in Nonlinear Functional Analysis,", Courant Institute of Mathematical Sciences, (1974).

[30]

P. H. Rabinowitz, Nonlinear Sturm-Liouville problems for second order ordinary differential equations,, Comm. Pure Appl. Math., 23 (1970), 939.

[31]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems,, J. Func. Anal., 7 (1971), 487.

[32]

P. H. Rabinowitz, A global theorem for nonlinear eigenvalue problems and applications,, Contributions to Nonlinear Functional Analysis, (1971), 11.

[33]

W. Radzki and S. Rybicki, Degenerate bifurcation points of periodic solutions of autonomous Hamiltonian systems,, J. Diff. Equat., 202 (2004), 284. doi: 10.1016/j.jde.2004.03.037.

[34]

H. Ruan and S. Rybicki, Applications of equivariant degree for gradient maps to symmetric Newtonian systems,, Nonl. Anal. TMA, 68 (2008), 1479. doi: 10.1016/j.na.2006.12.039.

[35]

K. Rybakowski, On the homotopy index for infinite dimensional semiflows,, Trans. Amer. Math. Soc., 269 (1982), 351. doi: 10.2307/1998452.

[36]

S. Rybicki, A degree for $S^1$-equivariant orthogonal maps and its applications to bifurcation theory,, Nonl. Anal. TMA, 23 (1994), 83. doi: 10.1016/0362-546X(94)90253-4.

[37]

S. Rybicki, Applications of degree for $S^1$-equivariant gradient maps to variational nonlinear problems with $S^1$-symmetries,, Topol. Meth. Nonl. Anal., 9 (1997), 383.

[38]

S. Rybicki, Degree for equivariant gradient maps,, Milan J. Math., 73 (2005), 103. doi: 10.1007/s00032-005-0040-2.

[39]

S. Rybicki, Global bifurcations of critical orbits via equivariant Conley index,, Adv. Nonl. Stud., 11 (2011), 929.

[40]

D. Salamon, Connected simple systems and the Conley index of isolated invariant sets,, TAMS, 291 (1985), 1. doi: 10.2307/1999893.

[41]

D. Salamon, Morse theory, the Conley index and Floer homology,, Bull. London Math. Soc., 22 (1990), 113. doi: 10.1112/blms/22.2.113.

[42]

J. Smoller and A. Wasserman, Bifurcation and symmetry-breaking,, Invent. Math., 100 (1990), 63. doi: 10.1007/BF01231181.

[43]

J. Smoller, "Shock Waves and Reaction-diffusion Equations,", Fundamental Principles of Mathematical Science, 258 (1983).

[44]

F. Takens, Some remarks on the Böhme-Berger bifurcation theorem,, Math. Z., 125 (1972), 359.

[45]

G. W. Whitehead, "Recent Advances in Homotopy Theory,", CMBS Regional Conf. Ser. in Math., 5 (1970).

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