The reduction principle for discrete dynamical and semidynamical systems in metric spaces.

*(English)*Zbl 0824.34049The 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.

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.

Reviewer: E.Elizalde (Barcelona)

##### 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 |

##### Keywords:

reduction principle; discrete dynamics; Hartman-Grobman theorem; autonomous differential equations; complete metric space
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\textit{A. Reinfelds}, Z. Angew. Math. Phys. 45, No. 6, 933--955 (1994; Zbl 0824.34049)

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##### References:

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