Embedding smooth and formal diffeomorphisms through the Jordan-Chevalley decomposition

[摘要] In [Xiang Zhang, The embedding flows of C-infinity 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. (C) 2012 Elsevier Inc. All rights reserved.

[发布日期] 2012-12-15

[关键词] Local diffeomorphism;Normal form;Embedding flow