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The topological behaviour of a diffeomorphism near a fixed point. (English) Zbl 0543.58021

Using an analytical technique along the line of J. Moser’s proof for the structural stability of Anosov systems, the author extends a topological linearization theorem for flows by K. Palmer to the case of maps. The main result is as follows.
Let \(\Phi (\Phi(x)=Mx+F(x)\), \(F(0)=0)\) be a Lipschitz-homeomorphism on a Banach space E, where \(M=diag\{M^ 0,M^-,M^+\}\in {\mathcal L}(E)\) is an isomorphism leaving the splitting \(E=E^ 0+E^-+E^+\) invariant, \(x=(x^ 0,x^-,x^+)\) and \(F=(F^ 0,F^-,F^+)\). The spectrum of \(M^-\) (respectively \(M^+)\) is inside (respectively outside) of the unit circle and, moreover, bounded away from the unit circle; \(M^ 0\) has its spectrum on the unit circle. Assume that the Lipschitz constant L(F) is sufficiently small. Then there is a unique invariant central manifold \(N=graph(H)\), given by \((x^-,x^+)=H(x^ 0)\), satisfying \(H(0)=0\), \(\sup | H|<\infty\) and \(L(H)=O(L(F))\). Moreover, the abovementioned homeomorphism \(\Phi\) is topologically equivalent to the following homeomorphism \(\Phi_ 0:x^ 0\!_ 1=M^ 0x^ 0+F^ 0(x^ 0,H(x^ 0)), x^-\!_ 1=M^-x^-, x^+\!_ 1=M^+x^+\).
Reviewer: T.Ding

MSC:

37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
37D99 Dynamical systems with hyperbolic behavior
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
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References:

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