WIAS Preprint No. 722, (2002)

Polynomial approximations of symplectic dynamics and richness of chaos in non-hyperbolic area-preserving maps



Authors

  • Turaev, Dmitry

2010 Mathematics Subject Classification

  • 37J10 37C15 37E30 37G25

Keywords

  • homoclinic tangency, Hénon map, standard map, Cremona group, Newhouse phenomenon

Abstract

It is shown that every symplectic map of $R^2n$ can be approximated, in the $C^infty$-topology, on any compact set, by some iteration of some map of the form $(x,y)mapsto (y+eta, -x +Phi(y))$ where $xin R^n$, $yin R^n$, and $Phi$ is a polynomial $R^nrightarrow R^n$ and $etain R^n$ is a constant vector. For the case of area-preserving maps (i.e. $n=1$), it is shown how this result can be applied to prove that $C^r$-universal maps (a map is universal if its iterations approximate dynamics of all $C^r$-smooth area-preserving maps altogether) are dense (in the $C^r$-topology) in the Newhouse regions.

Appeared in

  • Nonlinearity, 16 (2003), pp. 1-13

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