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The reduction principle for discrete dynamical and semidynamical systems in metric spaces. (English) Zbl 0824.34049
The author extends the results of the classical Hartman-Grobman theorem, which states that a system of autonomous differential equations of the form $$\dot x= Ax+ f(x, y)$$, $$\dot y= By+ g(x, y)$$, is dynamically equivalent to the linear system $$\dot x= Ax$$, $$\dot y= By$$, if the spectra of $$A$$ and $$B$$ are separated by the imaginary axis in the complex plane, and the mappings $$f$$ and $$g$$ are uniformly Lipschitzian with sufficiently small Lipschitz constant and vanish at the origin.
The author considers the case of a mapping $$T$$ in a complete metric space, defined by $$T(x, y)= (f(x, y), g(x, y))$$ and having a fixed point. He generalizes the results of A. Aulbach and B. M. Garay [Z. Angew. Math. Phys. 44, 469-494 (1993; Zbl 0803.46045), and 45, 505-542 (1994; Zbl 0813.47075)], concerning the decoupling and linearization for a mapping in a Banach space. By relaxing the hypothesis, he proves the more general statement that there are mappings $$u$$, $$v$$ and $$q$$ such that $$T$$ is topologically conjugate to the mapping $$R$$ defined by $$R(x, y)= (f(x, u(x)), g(q(x, y), y))$$. A theorem of conjugacy of a given, possibly non-invertible mapping which admits an invariant set, to a simpler mapping than the given one in terms of decoupling is proven in the paper too. The results are valid in an arbitrary complete metric space.

##### MSC:
 37-XX Dynamical systems and ergodic theory 39A10 Additive difference equations 34C20 Transformation and reduction of ordinary differential equations and systems, normal forms 34G20 Nonlinear differential equations in abstract spaces
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##### References:
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