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Perturbation of zeros in the presence of symmetries. (English) Zbl 0535.58013

In this paper, we consider the following situation. Assume that E is a finite-dimensional linear space, \(\{T_ g\}_{g\in G}\) is a linear representation of a compact Lie group G on E, and F:\(E\to E\) is G- invariant. Assume that \(F(k)=0\) and let \(M=\{T_ gk:g\in G\}\) (then \(F(x)=0\) for \(x\in M).Assume\) that F(x)\(\neq 0\) if x is near M and \(x\in \!| M\). We obtain necessary and sufficient conditions for every G- invariant mapping \(C^ 0\) close to F to have a zero near M (that is, the orbit M is G-stable for F). The condition involves a number of degrees and normalizer type conditions. We also show that, if F is \(C^ 1\), it may happen that M is not G-stable for F but every G-invariant mapping \(C^ 1\) close to F has a zero near M. This stability depends upon which norm we take the perturbation to be small. (More recently the author has proved the conjecture on p. 116 on stability to \(C^ 1\) small perturbations). The proofs depend upon a corrected version (which is proved in the paper) of a theorem of R. L. Rubinsztein [Dissertationes Math 134 (1976; Zbl 0343.57021)].
We also consider the case where F has a gradient structure and we look at perturbations with a gradient structure. In this case, normalized conditions do not occur. As sketched in the author’s paper [J. Reine Angew. Math. 350, 1-22 (1984; Zbl 0525.58012)], the question on p. 119 has a negative answer even if \(G=\{e\}\). In addition, the comment in Remark 1 on p. 6 of the author’s paper [loc. cit.] can be used to give a positive answer to the question on p. 118.

MSC:

58E07 Variational problems in abstract bifurcation theory in infinite-dimensional spaces
55Q70 Homotopy groups of special types
47J05 Equations involving nonlinear operators (general)
37C75 Stability theory for smooth dynamical systems
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