HÖLDER CONTINUITY AND LAMINARITY OF THE GREEN CURRENTS FOR HÉNON-LIKE MAPS

Abstract Under a natural assumption on the dynamical degrees, we prove that the Green currents associated to any Hénon-like map in any dimension have Hölder continuous super-potentials, i.e., give Hölder continuous linear functionals on suitable spaces of forms and currents. As a consequence, the unique measure of maximal entropy is the Monge-Ampère of a Hölder 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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