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arXiv · 1808.10130

Dynamics of holomorphic correspondences on Riemann Surfaces

Tien-Cuong Dinh·Lucas Kaufmann·Hao Wu
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Abstract

We study the dynamics of holomorphic correspondences $f$ on a compact Riemann surface $X$ in the case, so far not well understood, where $f$ and $f^{-1}$ have the same topological degree. Under a mild and necessary condition that we call non weak modularity, $f$ admits two canonical probability measures $\mu^+$ and $\mu^-$ which are invariant by $f^*$ and $f_*$ respectively. If the critical values of $f$ (resp. $f^{-1}$) are not periodic, the backward (resp. forward) orbit of any point $a \in X$ equidistributes towards $\mu^+$ (resp. $\mu^-$), uniformly in $a$ and exponentially fast.

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Tien-Cuong Dinh, Lucas Kaufmann, Hao Wu. 2018-08-30. Dynamics of holomorphic correspondences on Riemann Surfaces. https://arxiv.org/abs/1808.10130

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