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arXiv · 2101.12148

The Critical Locus and Rigidity of Foliations of Complex Henon Maps

Misha Lyubich·John W. Robertson
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Abstract

We study Henon maps which are perturbations of a hyperbolic polynomial p with connected Julia set. We give a complete description of the critical locus of these maps. In particular, we show that for each critical point c of p, there is a primary component of the critical locus asymptotic to the line y = c. Moreover, primary components are conformally equivalent to the punctured disk, and their orbits cover the whole critical set. We also describe the holonomy maps from such a component to itself along the leaves of two natural foliations. Finally, we show that a quadratic Henon map taken along with the natural pair of foliations, is a rigid object, in the sense that a conjugacy between two such maps respecting the foliations is a holomorphic or antiholomorphic affine map.

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Misha Lyubich, John W. Robertson. 2021-01-28. The Critical Locus and Rigidity of Foliations of Complex Henon Maps. https://arxiv.org/abs/2101.12148

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