Hyperbolicity is dense in the real quadratic family

TL;DR

This paper proves that hyperbolic maps are dense in the real quadratic family by showing topological and quasisymmetric conjugacy classes coincide for non-hyperbolic maps, leading to a uniqueness result via quasiconformal rigidity.

Contribution

It establishes the density of hyperbolic maps in the real quadratic family using conjugacy class analysis and quasiconformal rigidity.

Findings

Hyperbolic maps are dense in the real quadratic family.

Topological and quasisymmetric conjugacy classes coincide for non-hyperbolic maps.

Each conjugacy class has a unique representative in the quadratic family.

Abstract

It is shown that for non-hyperbolic real quadratic polynomials topological and quasisymmetric conjugacy classes are the same. By quasiconformal rigidity, each class has only one representative in the quadratic family, which proves that hyperbolic maps are dense.