On the postcritical set of a rational map

2017/09/20 by DeMarco, Laura G., Koch, Sarah C., McMullen, Curtis T. · 1 citation
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1709.06866

Abstract

The postcritical set P(f) of a rational map f:\mathbb P¹→ \mathbb P¹ is the smallest forward invariant subset of \mathbb P¹ that contains the critical values of f. In this paper we show that every finite set X⊂ \mathbb P¹(\mathbb Q) can be realized as the postcritical set of a rational map. We also show that every map F:X→ X defined on a finite set X⊂ \mathbb P¹(\mathbb C) can be realized by a rational map f:P(f)→ P(f), provided we allow small perturbations of the set X. The proofs involve Belyi's theorem and iteration on Teichmüller space.

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