The degree of proper \(C^ 2\) Fredholm mappings: Covariant theory
Let \(X\) be a Banach space equipped with a continuous action of a compact Lie group \(G\) and let \(f: X\to X\) be a \(G\)-covariant compact perturbation of the identity. Denote by \(X^G\) the fixed point space of \(G\) and by \({\mathcal J}^G\) the ideal of \(\mathbb{Z}\) generated by the integers \(\chi(G/ G_x)\), \(x\in X\setminus X^G\), \(\text{rank } G_x= \text{rank } G\), where \(G_x\) is the isotropy subgroup of \(x\) and \(\chi (G/ G_x)\) the Euler-Poincaré characteristic of \(G/ G_x\). Let \(\Omega \subset X\) be a \(G\)-invariant bounded open subset of \(X\) with \(0\not\in f(\partial \Omega)\). Then, it is known that \(\deg_{LS} (f, \Omega, 0)= \deg_{LS} (f|_{X^G}, \Omega \cap X^G, 0)\bmod {\mathcal J}^G\), where \(\deg_{LS}\) is Leray-Schauder's degree. This is a recent extension by P. J. Rabier of a result by \textit{Z. Wang} [Chin. Ann. Math., Ser. B 10, No. 4, 520-536 (1989; Zbl 0705.58006)]\ (where \(X\) is a Hilbert space and the action of \(G\) is smooth). The aim of this paper is to extend this result to covariant \(C^2\) Fredholm mappings \(f: X\to X\) in a real Banach space \(X\) and where the degree is that recently defined by the authors [J. Reine Angew. Math. 427, 1-33 (1992; Zbl 0745.47049)].
continuous action of a compact Lie group
\(G\)-covariant compact perturbation of the identity
scientific article; zbMATH DE number 683316