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

Equivariant Conley index versus degree for equivariant gradient maps

Anna Go??biewska 1,2,3, and  S?awomir Rybicki 1,2,3,

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.
Keywords: Equivariant Conley index, equivariant gradient degree, compact Lie group, strongly indefinite operator..
Mathematics Subject Classification: Primary: 37G40; Secondary: 58E0.
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[6]

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

[7]

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

[8]

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

[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.  Google Scholar

[10]

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

[11]

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

[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.  Google Scholar

[13]

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

[14]

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

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[19]

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

[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.  Google Scholar

[21]

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

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[35]

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

[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.  Google Scholar

[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.   Google Scholar

[38]

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

[39]

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

[40]

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

[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.  Google Scholar

[42]

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

[43]

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

[44]

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

[45]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[6]

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

[7]

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

[8]

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

[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.  Google Scholar

[10]

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

[11]

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

[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.  Google Scholar

[13]

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

[14]

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

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[19]

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

[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.  Google Scholar

[21]

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

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[35]

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

[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.  Google Scholar

[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.   Google Scholar

[38]

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

[39]

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

[40]

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

[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.  Google Scholar

[42]

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

[43]

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

[44]

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

[45]

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

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