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Invariant manifolds associated to invariant subspaces without invariant complements: a graph transform approach. (English) Zbl 1115.37019
Summary: We use the graph transform method to prove existence of invariant manifolds near fixed points of maps tangent to invariant subspaces of the linearization.
In contrast to the best known of such theorems, we do not assume that the corresponding space for the linear map is a spectral subspace. Indeed, we allow that there is no invariant complement (in particular, we do not need that the decomposition corresponds to spectral subspaces). We also do not need that the spectrum of the operator restricted to the spaces satisfies the usual dominance conditions.
We prove some uniqueness theorems and show how this can be used to prove results for flows.
More general theorems have been proved in Xavier CabrĂ©, Ernest Fontich and Rafael de la Llave [Indiana Univ. Math. J. 52, No. 2, 283–328 (2003; Zbl 1034.37016)] by another method.

MSC:
37D10 Invariant manifold theory for dynamical systems
37C10 Dynamics induced by flows and semiflows
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