Infinitesimal Thurston Rigidity and the Fatou-Shishikura Inequality

TL;DR

This paper refines the Fatou-Shishikura Inequality by leveraging Thurston's Rigidity Theorem, establishing a new bound on nonrepelling cycles based on critical orbit structure in rational maps.

Contribution

It introduces a novel proof of the inequality using Thurston's Rigidity Theorem and the injectivity of a specific operator on quadratic differentials.

Findings

Refined the Fatou-Shishikura Inequality

Connected Thurston's Rigidity with critical orbit analysis

Provided a new proof technique for the inequality

Abstract

We prove a refinement of the Fatou-Shishikura Inequality - that the total count of nonrepelling cycles of a rational map is less than or equal to the number of independent infinite forward critical orbits - from a suitable application of Thurston's Rigidity Theorem - the injectivity of $I-f_*$ on spaces of meromorphic quadratic differentials.