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Title:
On the accumulation of non-split separtrices by invariant circles
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Abstract:
A theorem by M.R. Herman,``Herman's last geometric theorem'', asserts that if a smooth orientation and area preserving diffeomorphism $f$ of the 2-plane $\R^2$ (or the 2-cylinder $\R/\Z\times\R$) admits a KAM curve $C$ (a smooth invariant curve on which the dynamics of $f$ is conjugated to a Diophantine translation) then $C$ is accumulated by other KAM curves, the union of which covers a set of positive 2-dimensional Lebesgue measure in any neighborhood of $C$. In this talk we shall investigate whether such a phenomenon holds if, instead of being a KAM circle, the invariant set $C$ is a (non-split) separatrix of a hyperbolic fixed point of $f$. This analysis might be useful for understanding symplectic diffeomorphisms with zero entropy or in the search of a smooth twist map admitting an isolated irrational invariant curve bounding two Birkhoff instability regions. The renormalization paradigm is an important element in our approach. This is a joint work with Anatole Katok.
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