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arXiv · 2606.08746

On the support of measures of large entropy for automorphisms of K\"ahler manifolds

Fabrizio Bianchi·Sobir Boymurodov·Karim Rakhimov
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Abstract

Let $f$ be a holomorphic automorphism of a compact K\"ahler manifold $X$ with simple action on cohomology. We show that every ergodic measure with sufficiently large entropy is supported on the Julia set of $f$. In particular, when $X$ is a surface, any ergodic measure with positive entropy is supported on the Julia set. The proof relies on quantitative estimates for the speed of convergence towards the Green currents of $f$, with respect to a suitable norm on an adapted functional space of non-necessarily closed currents.

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Fabrizio Bianchi, Sobir Boymurodov, Karim Rakhimov. 2026-06-07. On the support of measures of large entropy for automorphisms of K\"ahler manifolds. https://arxiv.org/abs/2606.08746

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