On hyperbolicity in the renormalization of near-critical area-preserving maps

Hans Koch

doi:10.3934/dcds.2016106

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2016, Volume 36, Issue 12: 7029-7056. Doi: 10.3934/dcds.2016106

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On hyperbolicity in the renormalization of near-critical area-preserving maps

Hans Koch^1,

1.
Department of Mathematics, The University of Texas at Austin, Austin, TX 78712

Received: January 31, 2016

Revised: May 31, 2016

Published: May 2017

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Abstract

We consider MacKay's renormalization operator for pairs of area-preserving maps, near the fixed point obtained in [1]. Of particular interest is the restriction $\mathfrak{R}_{0}$ of this operator to pairs that commute and have a zero Calabi invariant. We prove that a suitable extension of $\mathfrak{R}_{0}^{3}$ is hyperbolic at the fixed point, with a single expanding direction. The pairs in this direction are presumably commuting, but we currently have no proof for this. Our analysis yields rigorous bounds on various ``universal'' quantities, including the expanding eigenvalue.

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Mathematics Subject Classification: Primary: 37E20; Secondary: 37F25.

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