Semi-hyperbolic maps are rare

TL;DR

This paper proves that semi-hyperbolic rational maps are extremely rare in the space of all rational maps of a fixed degree, extending previous results from degree 2 to higher degrees.

Contribution

It generalizes earlier degree 2 results to all degrees at least 2, showing semi-hyperbolic maps have measure zero in the parameter space.

Findings

Semi-hyperbolic maps have Lebesgue measure zero

The result extends to all degrees d ≥ 2

Generalizes previous degree 2 case

Abstract

We prove in this paper that the set of semi-hyperbolic rational maps has Lebesgue measure zero in the space of rational maps of the Riemann sphere for a fixed degree d at least 2. It generalises an earlier result by J. Graczyk and the author who proved the same thing in degree 2.