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Lipschitz conjugacy of linear flows. (English) Zbl 1191.34060
Let $$\Phi(t,x)=\exp(At)x$$ and $$\Psi(t,x)=\exp(Bt)x$$ be the linear flows induced by autonomous linear systems $$x'=Ax$$ and $$x'=Bx$$ in a Euclidean space. It is well known that, in the hyperbolic case, the flows $$\Phi$$ and $$\Psi$$ are topologically conjugate if and only if the matrices $$A$$ and $$B$$ have the same number of eigenvalues with negative real parts. At the same time, the flows $$\Phi$$ and $$\Psi$$ are $$C^1$$-conjugate if and only if they are linearly conjugate.
The authors consider an intermediate case of Lipschitz conjugacy and show that the flows $$\Phi$$ and $$\Psi$$ are bi-Lipschitz conjugate if and only if the real Jordan forms of the matrices $$A$$ and $$B$$ coincide with the exception of simple Jordan blocks, where the imaginary parts of the eigenvalues may differ.

##### MSC:
 34C41 Equivalence and asymptotic equivalence of ordinary differential equations 34A30 Linear ordinary differential equations and systems, general
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