Abstract

In this paper we will give a short and elementary proof that critical relations unfold transversally in the space of rational maps.

Highlights

  • In this short paper we will give an elementary proof of some transversality properties for families of rational maps

  • The aim of this paper is to present a proof of transversality for rational maps with critical relations in a complete and readily accessible form

  • We are interested in the smoothness of sets defined by a set of critical relations of the form gm(ci (g)) = gn(c j (g))

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Summary

Introduction

In this short paper we will give an elementary proof of some transversality properties for families of rational maps. Our results hold in the setting of degenerate critical points and gives unfoldings of critical relations even when critical points share the same critical value. For this we use that Ratμd is a manifold and that Ratμd f → ( f (c1), . The idea of using quadratic differentials appeared first in Thurston’s characterization of post-critically finite branched covering of the 2-sphere [6] It has been used in for example [7, 15] and this was used in [33] to obtain a similar statement to ours for the quadratic case. The aim of this paper is to present a proof of transversality for rational maps with critical relations in a complete and readily accessible form.

Parametrising Rational Maps by Their Critical Values
Transversality Results for Rational Maps
How to Associate Critical Relations to a Rational Map
An Even More General Theorem
The Polynomial Case
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