Abstract
In order to establish some notation and terminology we recall that if f : ℝn → ℝn is continuous and Ω is an open bounded subset of ℝn such that f is different from 0 on the boundary of Ω, then there is defined an integer Deg(f, Ω) — the Brouwer (or topological) degree of f with respect to Ω. Obviously, if in the place of all continuos maps and all open bounded subsets of ℝn we take a smaller class of maps and/or a smaller class of subsets then we may try to define a topological invariant finer then the topological degree.
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References
H. Amann, A note on degree theory for gradient maps, Proc. AMS 85 (1982), 591–595.
T. Bartsch, Topological methods for variational problems with symmetries, Lecture Notes in Mathematics 1560, Springer-Verlag, Berlin, 1993.
G. E. Bredon, Introduction to Compact Transformation Groups, Academic Press, New York, 1972.
C. Conley, Isolated invarient sets and the Morse index, CBMS, Regional Confer. Ser. in Math., AMS, Providence, R.I., 1978.
C. Conley and E. Zehnder, A Morse type index theory for flows and periodic solutions to Hamiltonian systems, Comm. Pure Appi Math. 37 (1984), 207–253.
S.-N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer-Verlag, New-York, 1982.
E. N. Dancer, A new degree for 51-equivariant gradient mappings and applications, Ann. Inst. H. Poincaré, Anal. Non Linéaire 2 (1982), 329–370.
T. torn Dieck, Transformation Groups, de Gruyter, Berlin, 1987.
A. Dold, Lectures on Algebraic Topology, Springer-Verlag, New-York, 1972.
G. Dylawerski, K. Gęba, J. Jodel, and W. Marzantowicz, S 1-equivariant degree and the Puller index, Ann. Pol Math. 52 (1991), 243–280.
A. Floer, A refinement of the Conley index and an application to the stability of hyperbolic invariant sets, Ergod. Th. and Dynam. Sys. 7 (1987), 93–103.
M. W. Hirsch, Differential Topology, Springer-Verlag, New York, 1976.
K. Gęba, W. Krawcewicz, and J. Wu, An equivariant degree with applications to symmetric bifurcation problems. Part 1: Construction of the degree, Proc. London Math. Soc. 69 (1994), 377–398.
K. Gęba, I. Massabo, and A. Vignoli, On the Euler characteristic of equivariant gradient vector fields, Boll. Unione Mat. Italiana A.4(1990), 243–251.
J. Ize, I. Massabò, and A. Vignoli, Degree theory for equivariant maps I, Transaction AMS, 315 (1989), 433–510.
J. Ize, I. Massabò, and A. Vignoli, Degree theory for equivariant maps, the general S 1-action I, Memoirs AMS 100 (1992).
W. Krawceiwcz and P. Vivi, Generic bifurcations and equivariant degree, preprint.
W. Krawcewicz and J. Wu, Global bifurcation of time reversible systems, preprint.
N. Lloyd, Degree theory, Cambridge University Press, 1978.
J. Milnor, Topology from the Differentiable Viewpoint, The University Press of Virginia, Charlottesville, 1965.
J. Milnor, Lectures on the h-Cobordism Theorem, Princeton University Press, Princeton, 1965.
A. Parusiński, Gradient homotopies of gradient vector fields, Stadia Mathematica 46 (1990), 73–80.
G. Peschke, Degree of certain equivariant maps into a representation sphere, Topology Appl 59 (1994), 137–156.
P. H. Rabinowitz, A note on topological degree for potential operators, J. Math. Anal. Appl. 51 (1975), 483–492.
S. Rybicki, Applications of degree for S 1-equivariant gradient maps to variational nonlinear problems with S 1-symmetries, preprint.
S. Rybicki, On Rabinowitz alternative for the Laplace-Beltrami operator on S n −1. Continua that meet infinity, preprint.
D. Salamon, Connected simple systems and the Conley index of isolated invariant sets, Trans. Amer. Math. Soc. 291 (1985), 1–41.
A. Szymczak, a personal communication.
A. H. Wallace, Differential Topology, First Steps, Benjamin, New York, 1968.
Z-.Q. Wang, Symmetries, Morse polynomials and applications to bifurcation problems, Acta Mathematica Sinica, New Series 6 (1990), 165–177.
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Gęba, K. (1997). Degree for Gradient Equivariant Maps and Equivariant Conley Index. In: Matzeu, M., Vignoli, A. (eds) Topological Nonlinear Analysis II. Progress in Nonlinear Differential Equations and Their Applications, vol 27. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-4126-3_5
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DOI: https://doi.org/10.1007/978-1-4612-4126-3_5
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