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Controlling nonuniqueness of local invariant manifolds. (English) Zbl 0783.58061

Nonuniqueness of local invariant manifolds corresponding to certain spectral decompositions for \(A\) for the differential equation \[ \dot x=Ax+F(x),\quad x \in \mathbb{R}^ n, \] \(F\) being \(C^ r\) with \(F(0)=0\), \(DF(0)=0\), is discussed. It is proved that a unique \(C^ r\) local manifold can be selected by choosing its submanifold satisfying certain a priori estimates. The result is employed to prove for Morse-Smale systems a conjecture of M. Hirsch according to which certain Lipschitz continuous invariant manifolds of strongly cooperative systems are \(C^ 1\) smooth.

MSC:

37A30 Ergodic theorems, spectral theory, Markov operators
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
37D15 Morse-Smale systems
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