Journal of Modern Dynamics (JMD)

On cyclicity-one elliptic islands of the standard map

Pages: 153 - 208, Issue 2, June 2013      doi:10.3934/jmd.2013.7.153

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Jacopo De Simoi - Department of Mathematics, University of Toronto, 40 St George St., Toronto, ON M5S 2E4, Canada (email)

Abstract: We study the abundance of a special class of elliptic islands for the standard family of area-preserving diffeomorphism for large parameter values, i.e., far from the KAM regime. Outside a bounded set of parameter values, we prove that the measure of the set of parameter values for which an infinite number of such elliptic islands coexist is zero. On the other hand, we construct a positive Hausdorff dimension set of arbitrarily large parameter values for which the associated standard map admits infinitely many elliptic islands whose centers accumulate on a locally maximal hyperbolic set.

Keywords:  Standard Map, elliptic islands, conservative Newhouse phenomenon.
Mathematics Subject Classification:  Primary: 37D25; Secondary: 37E40, 37D45.

Received: March 2012;      Available Online: September 2013.