Abstract
A non-injective holomorphic self-cover of a Riemann surface induces a non-surjective holomorphic self-embedding of its Teichmüller space. We investigate the dynamics of such self-embeddings by applying our structure theorem of self-covering of Riemann surfaces and examine the distribution of its isometric vectors on the tangent bundle over the Teichmüller space. We also extend our observation to quasiregular self-covers of Riemann surfaces and give an answer to a certain problem on quasiconformal equivalence to a holomorphic self-cover.
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Fujikawa, E., Matsuzaki, K. & Taniguchi, M. Dynamics on Teichmüller spaces and self-covering of Riemann surfaces. Math. Z. 260, 865–888 (2008). https://doi.org/10.1007/s00209-008-0304-y
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DOI: https://doi.org/10.1007/s00209-008-0304-y