On the regularization of conservative maps. (English) Zbl 1211.37029

Avila, Artur

Acta Math. 205, No. 1, 5-18 (2010).

The 1936 Whitney Theorem says that \(C^{\infty}(M,N)\) maps are dense in \(C^r(M,N)\), \(r\in\mathbb N\cup\{0\}\). Palis and Pugh have been the first who asked about the corresponding regularization results in the case of conservative and symplectic maps. In 1976 Zender has proved the regularization theorem for symplectic structure and Zuppa has proved in 1979 the regularization theorem for conservative flows. The present paper implies the regularization theorem for conservative diffeomorphisms, resulting from the one for conservative flows by using results of Arbieto-Martheus, Dacorogna-Moser, Burago-Kleiner, McMullen.

Reviewer: Andrzej Piatkowski (Łódź)

MSC:

37D20	Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
37D25	Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
37D30	Partially hyperbolic systems and dominated splittings
37J10	Symplectic mappings, fixed points (dynamical systems) (MSC2010)
37C20	Generic properties, structural stability of dynamical systems

Keywords:

conservative map; symplectic diffeomorphism; partial hyperbolicity; Lyapunov exponent; generic properties

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