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arXiv · 1509.07105

On hyperbolic metric and asymptotically finite invariant differentials in holomorphic dynamics

Carlos Cabrera·Peter Makienko
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Abstract

Given a rational map $R$, we consider the complement of the postcritical set $S_R$. In this paper we discuss the existence of invariant Beltrami differentials supported on a $R$ invariant subset $A$ of $S_R$. Under some geometrical restrictions, either on the hyperbolic geometry of $A$ or on the asymptotic behavior of infinitesimal geodesics of the Teichm\"uller space of $S_R$, we show the absence of invariant Beltrami differentials supported on $A$. In particular, we show that if $A$ has finite hyperbolic area, then $A$ can not support invariant Beltrami differentials except in the case where $R$ is a Latt\`es map.

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Carlos Cabrera, Peter Makienko. 2015-09-23. On hyperbolic metric and asymptotically finite invariant differentials in holomorphic dynamics. https://arxiv.org/abs/1509.07105

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