The Collet-Eckmann condition for rational functions on the Riemann sphere (Q1944825)

A rational function \(R\) is said to satisfy the Collet-Eckmann condition, if there exists \(C,\gamma>0\) such that \[ |(R^n)'(R(c))|\geq C e^{\gamma n} \] for all \(n\geq 0\) and every critical point \(c\) in the Julia set of \(R\) whose forward orbit does not contain other critical points. The main result of the paper says that in the parameter space of rational maps of any fixed degree greater than 1, the set of maps satisfying the Collet-Eckmann condition has positive Lebesgue measure.