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arXiv · 2407.01984

H\"older continuity and laminarity of the Green currents for H\'enon-like maps

Fabrizio Bianchi·Tien-Cuong Dinh·Karim Rakhimov
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Abstract

Under a natural assumption on the dynamical degrees, we prove that the Green currents associated to any H\'enon-like map in any dimension have H\"older continuous super-potentials, i.e., give H\"older continuous linear functionals on suitable spaces of forms and currents. As a consequence, the unique measure of maximal entropy is the Monge-Amp\`ere of a H\"older continuous plurisubharmonic function and has strictly positive Hausdorff dimension. Under the same assumptions, we also prove that the Green currents are woven. When they are of bidegree $(1,1)$, they are laminar. In particular, our results generalize results known until now only in algebraic settings, or in dimension 2.

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Fabrizio Bianchi, Tien-Cuong Dinh, Karim Rakhimov. 2024-07-02. H\"older continuity and laminarity of the Green currents for H\'enon-like maps. https://doi.org/10.1017/s1474748025101588

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