Embedding smooth and formal diffeomorphisms through the Jordan-Chevalley decomposition |
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Authors: | Javier Ribón |
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Institution: | Instituto de Matemática, UFF, Rua Mário Santos Braga S/N Valonguinho, Niterói, Rio de Janeiro, 24020-140, Brazil |
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Abstract: | In Xiang Zhang, The embedding flows of C∞ hyperbolic diffeomorphisms, J. Differential Equations 250 (5) (2011) 2283-2298] Zhang proved that any local smooth hyperbolic diffeomorphism whose eigenvalues are weakly nonresonant is embedded in the flow of a smooth vector field. We present a new and more conceptual proof of such result using the Jordan-Chevalley decomposition in algebraic groups and the properties of the exponential operator.We characterize the hyperbolic smooth (resp. formal) diffeomorphisms that are embedded in a smooth (resp. formal) flow. We introduce a criterion showing that the presence of weak resonances for a diffeomorphism plus two natural conditions imply that it is not embeddable. This solves a conjecture of Zhang. The criterion is optimal, we provide a method to construct embeddable diffeomorphisms with weak resonances if we remove any of the conditions. |
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Keywords: | primary 34C20 37F75 secondary 34C41 34M25 |
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