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On Stochastic Sea of the Standard Map

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Abstract

Consider a generic one-parameter unfolding of a homoclinic tangency of an area preserving surface diffeomorphism. We show that for many parameters (residual subset in an open set approaching the critical value) the corresponding diffeomorphism has a transitive invariant set Ω of full Hausdorff dimension. The set Ω is a topological limit of hyperbolic sets and is accumulated by elliptic islands.

As an application we prove that a stochastic sea of the standard map has full Hausdorff dimension for sufficiently large topologically generic parameters.

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  1. Department of Mathematics, University of California, Irvine, CA, 92697, USA

    Anton Gorodetski

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Correspondence to Anton Gorodetski.

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Communicated by G. Gallavotti

This work was supported in part by NSF grants DMS–0901627 and IIS-1018433.

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Gorodetski, A. On Stochastic Sea of the Standard Map. Commun. Math. Phys. 309, 155–192 (2012). https://doi.org/10.1007/s00220-011-1365-z

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