Stable domains of automorphisms of Banach spaces at fixed points (Q2763920)

Let \(\phi\) be an automorphism (= biholomorphic and onto) of a normed space \(X\) with \(\phi (a)=a\), and let \(W(a,\phi)\) be the set \(\{x\in X\colon \lim_{k\to\infty}\phi^k(x)=a\}\), where \(\phi^1=\phi\) and \(\phi^k=\phi{\circ}\phi^{k-1}\) for \(k=2,3,\dots\). Also, let \(\Lambda_{\phi}(a)\) and \(\lambda_{\phi}(a)\) be, respectively, the supremum and the infimum of the set \(\{\|d\phi(a)(x)\|/\|x\|\colon x\in X, x\not=0\}\), where \(d\phi (a)\) is the linear part of \(\phi\) at \(a\). It is proved that if \(X\) is a Banach space and if \(\Lambda_{\phi}(a)^2<\lambda_{\phi}(a)\), then the sequence \(\{d\phi(a)^{-k}{\circ}\phi^k\}\) converges to a biholomorphic mapping \(f_{\phi}\) from \(W (a,\phi)\) onto \(X\), and the convergence is uniform on compact subsets of \(W(a,\phi)\).